International Mathematics Research Notices

Gonin,, Roman;Tsymbaliuk,, Alexander

doi: 10.1093/imrn/rnz083pmid: N/A

Abstract We propose a natural generalization of the construction of the quantum difference Toda lattice [6, 22] associated with a simple Lie algebra |$\mathfrak{g}$|⁠. Our construction depends on two orientations of the Dynkin diagram of |$\mathfrak{g}$| and some other data (which we refer to as a pair of Sevostyanov triples). In types |$A$| and |$C$|⁠, we provide an alternative construction via Lax matrix formalism, cf. [15]. We also show that the generating function of the pairing of Whittaker vectors in the Verma modules is an eigenfunction of the corresponding modified quantum difference Toda system and derive fermionic formulas for the former in spirit of [7]. We give a geometric interpretation of all Whittaker vectors in type |$A$| via line bundles on the Laumon moduli spaces and obtain an edge-weight path model for them, generalizing the construction of [4]. 1 Introduction In a recent work [10] of M. Finkelberg and the 2nd author, a family of |$3^{n-1}$| commutative subalgebras in the algebra of difference operators on |$({\mathbb{C}}^{\times })^{n+1}$| was constructed, generalizing the type |$A$| quantum difference Toda lattice of [6, 22]. In this paper, we show how the construction of [22] for an arbitrary semisimple Lie algebra |$\mathfrak{g}$| can be generalized to produce |$3^{\textrm{rk}(\mathfrak{g})-1}$|integrable systems, thus answering a question of P. Etingof. In types |$A$| and |$C$|⁠, we identify these systems with the ones obtained via the Lax matrix formalism. We also discuss some generalizations of the basic results on the quantum difference Toda system to the current setting. The importance of our generalization of |$q$|-Toda systems of [6, 22] is two-fold. First of all, as emphasized in [10] (historically this goes back at least to [20]) already, the quasi-classical limit of this construction (known as the relativistic open Toda system) crucially depends on a choice of a pair of Coxeter elements in the Weyl group of |$G$| (simply connected algebraic group associated with |$\mathfrak{g}$|⁠). One of our main results, Theorem 3.1, gives an upper bound on the number of different integrable systems we obtain this way in the quantum case. Another motivation arises from the geometric representation theory, where Whittaker vectors (closely related to the Toda systems due to Theorem 4.9) often have natural geometric interpretations that unveil additional symmetry. We illustrate this in Section 5, where the universal Verma module over |$U_v(\mathfrak{s}\mathfrak{l}_n)$| is realized as the equivariant |$K$|-theory of Laumon spaces due to [3] (see Theorem 5.3); one of the Whittaker vectors is realized as a sum of the structure sheaves (see (5.5) and Proposition 5.14(a)), while an extra symmetry noticed in Proposition 5.14(b) gives rise to a family of Whittaker vectors (see Theorem 5.5 and Proposition 5.17). This paper is organized as follows: In Section 2, we construct the modified quantum difference Toda systems depending on a pair of Sevostyanov triples (following [21, 22]) and generalizing the |$q$|-Toda systems of [6, 22]. In Section 3, we explain how to compute explicitly the corresponding hamiltonians using [13]. We write down the formulas for the hamiltonians corresponding to the 1st fundamental representation in the classical types and |$G_2$|⁠. In the latter case of |$G_2$|⁠, our formula seems to be new even in the simplest set-up of the standard |$q$|-Toda system of [6]; see (3.33).One of the key results of this section is that there are at most |$3^{\textrm{rk}(\mathfrak{g})-1}$| different modified quantum difference Toda systems; see Theorem 3.1. For the classical types and |$G_2$|⁠, see Theorem 3.2, whose proof is more elementary and relies on Propositions 3.11, 3.14, 3.17, 3.20, and 3.38. We also show that these are maximal commutative subalgebras, determined by their 1st hamiltonians; see Theorem 3.3.We also prove that in type |$A$| these integrable systems exactly match those of [10, 11(ii, iii)]; see Theorem 3.24. This generalizes the Lax matrix realization of the type |$A$||$q$|-Toda system, due to [15]. In Theorem 3.31, we also provide a similar Lax matrix realization of the type |$C$| modified quantum difference Toda systems. Noticing that the periodic counterparts of these two constructions in the classical case (i.e., for |$\vec{k}=\vec{0}$| in the notations of loc.cit.) match up with the hamiltonians of the affine |$q$|-Toda lattice of [6], see formulas (3.22, 3.29, 3.30, 3.31, 3.34), we propose a periodic analogue of the modified quantum difference Toda systems in types |$A, C$|⁠; see Propositions 3.26 and 3.33 and Remarks 3.28(b,c) and 3.35(b). In Section 4, we study the Shapovalov pairing between a pair of Whittaker vectors (determined by Sevostyanov triples) in Verma modules. We obtain fermionic formulas for those in spirit of [7]; see Theorems 4.6 and 4.7. We also prove that their generating function is naturally an eigenfunction of the corresponding modified quantum difference Toda system; see Theorem 4.9. In Section 5, we provide a geometric interpretation of all type |$A$| Whittaker vectors and their Shapovalov pairing via the geometry of the Laumon moduli spaces, generalizing [3]; see Theorems 5.5 and 5.11.Following a suggestion of B. Feigin, we relate this family of Whittaker vectors to an eigen-property of the (geometrically) simplest one (5.5) with respect to the action of the quantum loop algebra |$U_v(L\mathfrak{s}\mathfrak{l}_n)$| (via the evaluation homomorphism); see Propositions 5.14 and 5.17 and Corollary 5.16. This viewpoint also provides an edge-weight path model for a general type |$A$| Whittaker vector, generalizing the path model of [4] for a particular choice of a Sevostyanov triple; see Propositions 5.19 and 5.21. In Appendices, we prove Proposition 3.11 and Theorems 3.1, 3.2, 3.3, and 3.24. 2 Sevostyanov Triples and Whittaker Functions 2.1 Quantum groups We fix the notations as follows. Let |$G$| be a simply connected complex algebraic group with a semisimple Lie algebra |$\mathfrak{g}$|⁠. We denote by |$H\subset B$| a pair of a Cartan torus and a Borel subgroup. The Cartan subalgebra |$\mathfrak{h}\subset \mathfrak{g}$| is defined as the Lie algebra of |$H$|⁠, |$\Delta $| denotes the set of roots of |$(\mathfrak{g},\mathfrak{h})$|⁠, and |$\Delta _+\subset \Delta $| denotes the set of positive roots corresponding to |$B$|⁠. Let |$n=\textrm{rk}(\mathfrak{g})$| be the rank of |$\mathfrak{g}$|⁠, |$\alpha _1,\ldots ,\alpha _n$| be the simple positive roots, and |$\omega _1,\ldots ,\omega _n$| be the fundamental weights. Let |$P:=\oplus _{i=1}^n {\mathbb{Z}}\omega _i$| be the weight lattice, |$Q:=\oplus _{i=1}^n {\mathbb{Z}}\alpha _i$| be the root lattice, and set |$P_+:=\oplus _{i=1}^n {\mathbb{Z}}_{\geq 0}\omega _i,\ Q_+:=\oplus _{i=1}^n {\mathbb{Z}}_{\geq 0}\alpha _i$|⁠. We write |$\beta \geq \gamma $| if |$\beta -\gamma \in Q_+$|⁠. We fix a nondegenerate invariant bilinear form |$(\cdot ,\cdot )\colon \mathfrak{h}\times \mathfrak{h}\to{\mathbb{C}}$| and identify |$\mathfrak{h}^*$| with |$\mathfrak{h}$| via |$(\cdot ,\cdot )$|⁠. We set |${\mathsf{d}}_i:=\frac{(\alpha _i,\alpha _i)}{2}$|⁠. The choice of |$(\cdot ,\cdot )$| is such that |${\mathsf{d}}_i=1$| for short roots |$\alpha _i$|⁠, in particular, |${\mathsf{d}}_i\in \{1,2,3\}$| for any |$i$|⁠. We also define |$\omega _i^{\vee }:=\omega _i/{\mathsf{d}}_i$| so that |$(\omega _i^{\vee },\alpha _j)=\delta _{i,j}$|⁠, and |$\rho :=\sum _{i=1}^n \omega _i=\frac{1}{2}\sum _{\gamma \in \Delta _+}\gamma \in P$|⁠. Let |$(a_{ij})_{i,j=1}^n$| be the corresponding Cartan matrix with |$a_{ij}=\frac{2(\alpha _i,\alpha _j)}{(\alpha _i,\alpha _i)}$|⁠. We define |$b_{ij}={\mathsf{d}}_ia_{ij}=(\alpha _i,\alpha _j)$|⁠, so that |$(b_{ij})_{i,j=1}^n$| is symmetric. Choose |$\mathsf{N}\in{\mathbb{Z}}_{>0}$| so that |$(P,P)\subset \frac{1}{\mathsf{N}}{\mathbb{Z}}$|⁠. The quantum group (of adjoint type in the terminology of [16]) |$U_v(\mathfrak{g})$| is the unital associative |${\mathbb{C}}(v^{1/\mathsf{N}})$|-algebra generated by |$\{E_i,F_i,K_{\mu }\}_{1\leq i\leq n}^{\mu \in P}$| with the following defining relations: \begin{align*} & K_{\mu} K_{\mu^{\prime}}=K_{\mu+\mu^{\prime}},\ K_0=1,\\ & K_{\mu} E_i K^{-1}_{\mu}=v^{(\mu,\alpha_i)}E_i,\ K_{\mu} F_i K^{-1}_{\mu}=v^{-(\mu,\alpha_i)}F_i,\ [E_i,F_j]=\delta_{i,j}\frac{K_i-K_i^{-1}}{v_i-v_i^{-1}},\\ & \sum_{r=0}^{1-a_{ij}}(-1)^r {1-a_{ij}\brack r}_{v_i} E_i^{1-a_{ij}-r}E_jE_i^{r}=0,\ \sum_{r=0}^{1-a_{ij}}(-1)^r {1-a_{ij}\brack r}_{v_i} F_i^{1-a_{ij}-r}F_jF_i^{r}=0\ (i\ne j), \end{align*} where |$K_i:=K_{\alpha _i},\ v_i:=v^{{\mathsf{d}}_i},\ [r]_v:=\frac{v^r-v^{-r}}{v-v^{-1}},\ [r]_v!:=[1]_v\cdots [r]_v,\ {m\brack r}_v:=\frac{[m]_v!}{[r]_v!\cdot [m-r]_v!}$|⁠. Set |$L_i:=K_{\omega _i}$|⁠. Since |$P=\oplus _{i=1}^n {\mathbb{Z}}\omega _i$|⁠, we will alternatively view |$U_v(\mathfrak{g})$| as the |${\mathbb{C}}(v^{1/\mathsf{N}})$|-algebra generated by |$\{E_i,F_i,L_i^{\pm 1}\}_{i=1}^n$| with the corresponding defining relations. In particular, \begin{equation*} L_iE_jL_i^{-1}=v_i^{\delta_{i,j}}E_j,\ L_iF_jL_i^{-1}=v_i^{-\delta_{i,j}}F_j,\ K_i=\prod_{j=1}^n L_j^{a_{ji}}. \end{equation*} 2.2 Sevostyanov triples Let |$\textrm{Dyn}(\mathfrak{g})$| be the graph obtained from the Dynkin diagram of |$\mathfrak{g}$| by replacing all multiple edges by simple ones, for example, |$\textrm{Dyn}(\mathfrak{s}\mathfrak{p}_{2n})=\textrm{Dyn}(\mathfrak{s}\mathfrak{o}_{2n+1})=\textrm{Dyn}(\mathfrak{s}\mathfrak{l}_{n+1})=A_n$|⁠. Given an orientation |$\textrm{Or}$| of |$\textrm{Dyn}(\mathfrak{g})$|⁠, define the associated matrix |$\epsilon =(\epsilon _{ij})_{i,j=1}^n$| via \begin{equation*} \epsilon_{ij}= \begin{cases} 0, & \textrm{if}\ a_{ij}=0\ \textrm{or}\ i=j,\\ 1, & \textrm{if}\ a_{ij}<0\ \textrm{and the edge is oriented} i \rightarrow j \textrm{in}\ \textrm{Or},\\ -1, & \textrm{if}\ a_{ij}<0\ \textrm{and the edge is oriented}\ i \leftarrow j \ \textrm{in}\ \textrm{Or}. \end{cases} \end{equation*} Definition 2.3. A Sevostyanov triple is a collection of the following data: (a) an orientation |$\textrm{Or}$| of |$\textrm{Dyn}(\mathfrak{g})$|⁠, (b) an integer matrix |$\mathsf{n}=(\mathsf{n}_{ij})_{i,j=1}^n$| satisfying |${\mathsf{d}}_j\mathsf{n}_{ij}-{\mathsf{d}}_i\mathsf{n}_{ji}=\epsilon _{ij}b_{ij}$| for any |$i,j$|⁠, (c) a collection |$c=(c_i)_{i=1}^n\in ({\mathbb{C}}(v^{1/\mathsf{N}})^{\times })^n$|⁠. We refer to this Sevostyanov triple by |$(\epsilon , \mathsf{n}, c)$|⁠. Fix a pair of integer matrices |$\mathsf{n}^{\pm }=(\mathsf{n}^{\pm }_{ij})_{i,j=1}^n$| and collections |$c^{\pm }=(c^{\pm }_i)_{i=1}^n\in ({\mathbb{C}}(v^{1/\mathsf{N}})^{\times })^n$|⁠. Set |$e_i:=E_i\cdot \prod _{p=1}^n L_p^{\mathsf{n}^+_{ip}},\ f_i:=\prod _{p=1}^n L_p^{-\mathsf{n}^-_{ip}}\cdot F_i$|⁠, and let |$U^{+}_{\mathsf{n}^+}(\mathfrak{g}), U^{-}_{\mathsf{n}^-}(\mathfrak{g})$| be the |${\mathbb{C}}(v^{1/\mathsf{N}})$|-subalgebras of |$U_v(\mathfrak{g})$| generated by |$\{e_i\}_{i=1}^n$| and |$\{f_i\}_{i=1}^n$|⁠, respectively. The following simple, but very important, observation is essentially due to [21]. Lemma 2.4. (a) The assignment |$e_i\mapsto c^+_i\ (1\leq i\leq n)$| extends to an algebra homomorphism |$\chi ^+\colon U^{+}_{\mathsf{n}^+}(\mathfrak{g})\to{\mathbb{C}}(v^{1/\mathsf{N}})$| if and only if there exists an orientation |$\textrm{Or}^+$| of |$\textrm{Dyn}(\mathfrak{g})$| with an associated matrix |$\epsilon ^+$|⁠, such that |$(\epsilon ^+,\mathsf{n}^+,c^+)$| is a Sevostyanov triple. (b) The assignment |$f_i\mapsto c^-_i\ (1\leq i\leq n)$| extends to an algebra homomorphism |$\chi ^-\colon U^{-}_{\mathsf{n}^-}(\mathfrak{g})\to{\mathbb{C}}(v^{1/\mathsf{N}})$| if and only if there exists an orientation |$\textrm{Or}^-$| of |$\textrm{Dyn}(\mathfrak{g})$| with an associated matrix |$\epsilon ^-$|⁠, such that |$(\epsilon ^-,\mathsf{n}^-,c^-)$| is a Sevostyanov triple. Proof. (a) As the “if” part is proved in [21, Theorem 4], let us now prove the “only if” part following similar arguments. Due to the triangular decomposition of |$U_v(\mathfrak{g})$|⁠, the algebra |$U^{+}_{\mathsf{n}^+}(\mathfrak{g})$| is generated by |$\{e_i\}_{i=1}^n$| subject to |$\sum _{r=0}^{1-a_{ij}}(-1)^rv^{r({\mathsf{d}}_j\mathsf{n}^+_{ij}-{\mathsf{d}}_i\mathsf{n}^+_{ji})} {1-a_{ij}\brack r}_{v_i} e_i^{1-a_{ij}-r}e_je_i^{r}=0$| for |$i\ne j$|⁠. Hence, there is a character |$\chi ^+\colon U^{+}_{\mathsf{n}^+}(\mathfrak{g})\to{\mathbb{C}}(v^{1/\mathsf{N}})$| with |$\chi ^+(e_i)\ne 0$| if and only if |$\sum _{r=0}^{1-a_{ij}}(-1)^rv^{r({\mathsf{d}}_j\mathsf{n}^+_{ij}-{\mathsf{d}}_i\mathsf{n}^+_{ji})} {1-a_{ij}\brack r}_{v_i}=0$| for any |$i\ne j$|⁠. If |$a_{ij}=0$|⁠, then we immediately get |${\mathsf{d}}_j\mathsf{n}^+_{ij}-{\mathsf{d}}_i\mathsf{n}^+_{ji}=0$|⁠. If |$a_{ij}=-1$|⁠, then we recover |${\mathsf{d}}_j\mathsf{n}^+_{ij}-{\mathsf{d}}_i\mathsf{n}^+_{ji}\in \{\pm{\mathsf{d}}_i\}=\{\pm b_{ij}\}$| and |$\epsilon _{ij}\in \{\pm 1\}$|⁠. Finally, if |$a_{ij}<-1$|⁠, then |$a_{ji}=-1$| and we can apply the previous case. (b) Analogous. 2.5 Whittaker functions From now on, we fix a pair of Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$|⁠, which give rise to the subalgebras |$U^{\pm }_{\mathsf{n}^{\pm }}(\mathfrak{g})$| of |$U_v(\mathfrak{g})$| and the corresponding characters |$\chi ^{\pm }\colon U^{\pm }_{\mathsf{n}^{\pm }}(\mathfrak{g})\to{\mathbb{C}}(v^{1/\mathsf{N}})$| of Lemma 2.4. We consider the quantum function algebra |${\mathcal{O}}_v(G)$| spanned by the matrix coefficients of integrable |$U_v(\mathfrak{g})$|-modules (with the highest weights in |$P_+$|⁠). Let |${\mathcal D}_v(G)$| denote the corresponding Heisenberg double [19, Section 3]. It acts on |${\mathcal{O}}_v(G)$|⁠. It is equipped with a homomorphism |$\mu _v\colon U_v(\mathfrak{g})\otimes U_v(\mathfrak{g})\to{\mathcal D}_v(G)$|⁠. Let |${\mathcal{O}}_v(Bw_0B)$| stand for the quantized coordinate ring of the big Bruhat cell [11, 8.2] (a localization of |${\mathcal{O}}_v(G)$|⁠). The action of |${\mathcal D}_v(G)$| on |${\mathcal{O}}_v(G)$| extends to the action on |${\mathcal{O}}_v(Bw_0B)$|⁠. In particular, |$U^-_{\mathsf{n}^-}(\mathfrak{g})\otimes U^+_{\mathsf{n}^+}(\mathfrak{g})\subset U_v(\mathfrak{g})\otimes U_v(\mathfrak{g})$| acts on |${\mathcal{O}}_v(Bw_0B)$|⁠. According to [11, (3.22), Theorem 4.7, Proposition 8.3], there are subalgebras |$S^{\pm }_v$| of |${\mathcal{O}}_v(Bw_0B)$| (we note that |${\mathcal{O}}_v(G), {\mathcal{O}}_v(Bw_0B),S^+_v$| are denoted by |$R_v[G], R_v[Bw_0B],\overline{S}^{\mp }_{w_0}$|⁠, respectively, in [11]) such that |${\mathcal{O}}_v(Bw_0B)\simeq S^-_v\otimes{\mathcal{O}}_v(H)\otimes S^+_v$| (as vector spaces) and |$S^{\pm }_v\simeq U^{\pm }_v(\mathfrak{g})$|⁠, where |$U^-_v(\mathfrak{g}), U^+_v(\mathfrak{g})$| are the subalgebras of |$U_v(\mathfrak{g})$| generated by |$\{F_i\}_{i=1}^n$| and |$\{E_i\}_{i=1}^n$|⁠, respectively. Hence, there is an (vector space) isomorphism \begin{equation} {\mathcal{O}}_v(Bw_0B)\simeq U^-_{\mathsf{n}^-}(\mathfrak{g})\otimes{\mathcal{O}}_v(H)\otimes U^+_{\mathsf{n}^+}(\mathfrak{g}), \end{equation} (2.1) under which the above actions of |$U^-_{\mathsf{n}^-}(\mathfrak{g}),U^+_{\mathsf{n}^+}(\mathfrak{g})$| on |${\mathcal{O}}_v(Bw_0B)$| are via the left and the right multiplications. Let |$U^{\pm }_{\mathsf{n}^{\pm }}(\mathfrak{g})^{\wedge }$| denote the completions of |$U^{\pm }_{\mathsf{n}^{\pm }}(\mathfrak{g})$| with respect to the natural gradings with |$\deg (e_i)=1$| and |$\deg (f_i)=1$|⁠. In view of the identification (2.1), we define the completion of |${\mathcal{O}}_v(Bw_0B)$| via |${\mathcal{O}}_v(Bw_0B)^{\wedge }\simeq U^-_{\mathsf{n}^-}(\mathfrak{g})^{\wedge }\otimes{\mathcal{O}}_v(H)\otimes U^+_{\mathsf{n}^+}(\mathfrak{g})^{\wedge }$|⁠. Hence, the subspace of semi-invariants |$\left ({\mathcal{O}}_v(Bw_0B)^{\wedge }\right )^{U^-_{\mathsf{n}^-}(\mathfrak{g})\otimes U^+_{\mathsf{n}^+}(\mathfrak{g}),\chi ^-\otimes \chi ^+}$| projects isomorphically onto |${\mathcal{O}}_v(H)$| under the restriction projection |${\mathcal{O}}_v(Bw_0B)^{\wedge }\to{\mathcal{O}}_v(H)$|⁠. We denote this projection by |$\phi \mapsto \phi _{\mid _H}$|⁠. Definition 2.6. A Whittaker function is an element of |$\left ({\mathcal{O}}_v(Bw_0B)^{\wedge }\right )^{U^-_{\mathsf{n}^-}(\mathfrak{g})\otimes U^+_{\mathsf{n}^+}(\mathfrak{g}),\chi ^-\otimes \chi ^+}$|⁠. Remark 2.7. Following [6], we could alternatively work with the dual quantum formal group |${\mathcal{A}}_{\hbar }(\mathfrak{g})=U_{\hbar }(\mathfrak{g})^*$|⁠, defined as the space of linear functions on |$U_{\hbar }(\mathfrak{g})$|⁠. Here the quantum group |$U_{\hbar }(\mathfrak{g})$| is defined over |${\mathbb{C}}[[\hbar ]]$| with |$v$| replaced by |$e^{\hbar }$|⁠. In this set-up, a Whittaker function is an element |$\phi \in{\mathcal{A}}_{\hbar }(\mathfrak{g})$| such that |$\phi (x^-xx^+)=\chi ^-(x^-)\chi ^+(x^+)\phi (x)$| for any |$x^{\pm } \in U^{\pm }_{\mathsf{n}^{\pm }}(\mathfrak{g}), x\in U_{\hbar }(\mathfrak{g})$|⁠. Let us point out that this differs from the notion of Whittaker functions as defined in loc.cit. We note that the character lattice |$X^*(H)=P$| and the pairing |$(Q,P)\subset{\mathbb{Z}}$|⁠, hence, we have the natural embedding of |$Q$| into the cocharacter lattice |$X_*(H)$|⁠. Thus, for every |$\lambda \in Q$| we can define the difference operators |$T_{\lambda }$| acting on |${\mathcal{O}}_v(H)$| via |$(T_{\lambda } f)(x)=f(x\cdot v^{\lambda })$|⁠. Moreover, since |$v^{(P,P)}\subset{\mathbb{C}}(v^{1/\mathsf{N}})$|⁠, the difference operators |$T_{\lambda }$| are also well-defined for |$\lambda \in P$|⁠. Let |$\widetilde{\mathcal{D}}_v(H)$| be the algebra generated by |$\{e^{\lambda },T_{\mu }|\lambda ,\mu \in P\}$|⁠, and |${\mathcal{D}}_v(H^{\textrm{ad}})$| be its subalgebra generated by |$\{e^{\lambda },T_{\mu }|\lambda \in Q,\mu \in P\}$|⁠. The following is completely analogous to [6, Proposition 3.2]. Lemma 2.8. (a) For any |$Y\in U_v(\mathfrak{g})$|⁠, there exists a unique difference operator |$\tilde{\textbf{D}}_Y=\tilde{\textbf{D}}_Y(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })\in \widetilde{{\mathcal{D}}}_v(H)$| such that |$(Y\phi )_{\mid _H}=\tilde{\textbf{D}}_Y(\phi _{\mid _H})$| for any Whittaker function |$\phi $|⁠. (b) |$\tilde{\textbf{D}}_Y$| is an element of |${\mathcal{D}}_v(H^{\textrm{ad}})\subset \widetilde{{\mathcal{D}}}_v(H)$|⁠. (c) If |$Y_1$| and |$Y_2$| are central elements of |$U_v(\mathfrak{g})$|⁠, then |$\tilde{\textbf{D}}_{Y_1Y_2}=\tilde{\textbf{D}}_{Y_1}\tilde{\textbf{D}}_{Y_2}$|⁠. Recall the element |$\Theta \in (U_v(\mathfrak{g})\otimes U_v(\mathfrak{g}))^{\wedge }$| of the completion of the vector space |$U_v(\mathfrak{g})\otimes U_v(\mathfrak{g})$| as defined in [16, 4.1.1]. Loosely speaking, the universal |$R$|-matrix is given by |$R=\Theta ^{\textrm{op}} \cdot R^0$|⁠, where |$R^0=v^T$| and |$T\in \mathfrak{h}\otimes \mathfrak{h}$| stand for the canonical element. Let |$\pi _V\colon U_v(\mathfrak{g})\to \textrm{End}(V)$| be a finite-dimensional representation, |$\{w_k\}_{k=1}^N$| be a weight basis of |$V$|⁠, and |$\mu _k\in P$| be the weight of |$w_k$|⁠. First, we note that though |$\Theta ,\Theta ^{\textrm{op}}$| are defined as infinite sums, their images |$(\textrm{id}\otimes \pi _V)(\Theta ), (\textrm{id}\otimes \pi _V)(\Theta ^{\textrm{op}})\in U_v(\mathfrak{g})\otimes \textrm{End}(V)$| are well-defined. Second, the image |$(\textrm{id}\otimes \pi _V)(R^0)=(\textrm{id}\otimes \pi _V)((R^0)^{\textrm{op}})\in U_v(\mathfrak{g})\otimes \textrm{End}(V)$| is also well-defined via |$(\textrm{id}\otimes \pi _V)(R^0)=\sum _{k=1}^N K_{\mu _k}\otimes E_{k,k}$| with |$E_{k,k}\in \textrm{End}(V)$| given by |$E_{k,k}(w_{k^{\prime}})=\delta _{k,k^{\prime}}w_{k^{\prime}}$| (this does not depend on the choice of a weight basis |$\{w_k\}$|⁠). Hence, working over |${\mathbb{C}}(v^{1/\mathsf{N}})$| (rather than in the formal setting |${\mathbb{C}}[[\hbar ]]$| as in [6, 22]), the elements |$(\textrm{id}\otimes \pi _V)(R)$| and |$(\textrm{id}\otimes \pi _V)(R^{\textrm{op}})$| are still well-defined. Due to [5, 18], the center of |$U_v(\mathfrak{g})$| is spanned by elements |$C_V$| corresponding to finite-dimensional |$U_v(\mathfrak{g})$|-representations |$V$| via the formula \begin{equation} C_V=\textrm{tr}_V(\textrm{id}\otimes \pi_V)\left(R^{\textrm{op}} R(1\otimes K_{2\rho})\right). \end{equation} (2.2) We define |$\tilde{\textbf{D}}_V, \textbf{D}_V\in{\mathcal{D}}_v(H^{\textrm{ad}})$| via |$\tilde{\textbf{D}}_V:=\tilde{\textbf{D}}_{C_V}$| and |$\textbf{D}_V:=e^{\rho }\tilde{\textbf{D}}_Ve^{-\rho }$|⁠. Consider the fundamental representations |$\{V_i\}_{i=1}^n$| of |$U_v(\mathfrak{g})$| and set |$\tilde{\textbf{D}}_i:=\tilde{\textbf{D}}_{V_i}, \textbf{D}_i:=\textbf{D}_{V_i}$|⁠. According to Lemma 2.8, |$\{\tilde{\textbf{D}}_i\}_{i=1}^n$| and therefore |$\{\textbf{D}_i\}_{i=1}^n$| are families of pairwise commuting elements of |${\mathcal{D}}_v(H^{\textrm{ad}})$|⁠. Definition 2.9. A modified quantum difference Toda system is the commutative subalgebra |${\mathcal{T}}={\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| of |${\mathcal{D}}_v(H^{\textrm{ad}})$| generated by |$\{\textbf{D}_i\}_{i=1}^n$|⁠. Due to Theorem 3.3(c) below, |$\textbf{D}_V\in{\mathcal{T}}$| for any finite-dimensional |$U_v(\mathfrak{g})$|-representation |$V$|⁠. Remark 2.10. This construction is a |$q$|-deformed version of the Kazhdan–Kostant approach to the classical Toda system. In case the two Sevostyanov triples coincide, we recover the original construction of [22]. Let us point out right away that we do not know how to generalize an alternative approach of [6] to obtain our modified quantum difference Toda systems. 3 1st Hamiltonians, Classification, and Lax Realization in Types A,C The main result of this section is the following. Theorem 3.1. There are at most |$3^{n-1}$| different modified quantum difference Toda systems, up to algebra automorphisms of |${\mathcal{D}}_v(H^{\textrm{ad}})$|⁠. The proof of this result is presented in Appendix D and crucially relies on Theorem 4.7. We also provide a more straightforward proof for the classical types |$A_n,B_n,C_n,D_n$| as well as the exceptional type |$G_2$|⁠. To state the result, we label the simple roots |$\{\alpha _i\}_{i=1}^n$| as in [2, Chapter VI, Section 4] (here |$n=2$| for the type |$G_2$|⁠). Given a pair of Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$|⁠, we define |$\vec{\epsilon }=(\epsilon _{n-1},\ldots ,\epsilon _1)\in \{-1,0,1\}^{n-1}$| via $$\epsilon _i:=\begin{cases} \frac{\epsilon ^+_{n-2,n}-\epsilon ^-_{n-2,n}}{2}, & \textrm{if}\ i=n-1\ \textrm{in type}\ D_n,\\ \frac{\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}}{2}, & \textrm{otherwise}. \end{cases}$$ Theorem 3.2. If |$\mathfrak{g}$| is of type |$A_n,B_n,C_n,D_n$| or |$G_2$|⁠, then up to algebra automorphisms of |${\mathcal{D}}_v(H^{\textrm{ad}})$|⁠, the modified quantum difference Toda system |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| depends only on |$\vec{\epsilon }$|⁠. We present the proof of this result in Appendix B. The key ingredient in our proof is that the 1st hamiltonian |$\textbf{D}_1$| depends only on |$\vec{\epsilon }\in \{-1,0,1\}^{n-1}$| up to an algebra automorphism of |${\mathcal{D}}_v(H^{\textrm{ad}})$|⁠, which is established case-by-case in Propositions 3.11, 3.14, 3.17, 3.20, and 3.38. Following an elegant argument of P. Etingof, we show in Appendix B that the other hamiltonians |$\textbf{D}_i$| match as well under the same automorphism. Let |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$| be the subalgebra of |${\mathcal{D}}_v(H^{\textrm{ad}})$|⁠, generated by |$\{e^{-\alpha _i},T_{\mu }\}_{1\leq i\leq n}^{\mu \in P}$|⁠. It follows from the construction that |$\textbf{D}_i\in{\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$|⁠, so that |${\mathcal{T}}\subset{\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$|⁠. Applying ideas similar to those from the proof of Theorem 3.2, we get another important result. Theorem 3.3. Consider a modified quantum difference Toda system |${\mathcal{T}}={\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$|⁠. (a) The difference operators |$\{\textbf{D}_i\}_{i=1}^n\subset{\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| are algebraically independent. (b) The centralizer of |$\textbf{D}_1$| in |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$| coincides with |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$|⁠. (c) We have |$\textbf{D}_V(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })\in{\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| for any finite-dimensional |$U_v(\mathfrak{g})$|-module |$V$|⁠. The proof of Theorem 3.3 is presented in Appendix C. 3.4 R-matrix and convex orderings Our computations are based on the explicit formula for the universal |$R$|-matrix |$R$|⁠, due to [13]. First, let us recall the construction of Cartan–Weyl root elements |$\{E_{\gamma },F_{\gamma }\}_{\gamma \in \Delta _+}$|⁠, which is crucially based on the notion of a convex ordering on |$\Delta _+$|⁠. Definition 3.5. An ordering |$\prec $| on the set of positive roots |$\Delta _+$| is called convex (we note that such orderings are called normal in [13, 23]) if for any three roots |$\alpha ,\beta ,\gamma \in \Delta _+$| such that |$\gamma =\alpha +\beta $|⁠, we have either |$\alpha \prec \gamma \prec \beta $| or |$\beta \prec \gamma \prec \alpha $|⁠. Fix a convex ordering |$\prec $| on |$\Delta _+$|⁠. For a simple root |$\alpha _i\ (1\leq i\leq n)$|⁠, set |$E_{\alpha _i}:=E_i, F_{\alpha _i}:=F_i$|⁠. To construct the remaining root vectors, we apply the following inductive algorithm. Let |$\alpha ,\beta ,\gamma \in \Delta _+$| be such that |$\gamma =\alpha +\beta , \alpha \prec \beta $|⁠, and there are no |$\alpha \npreceq \alpha ^{\prime}\prec \beta ^{\prime}\npreceq \beta $| satisfying |$\gamma =\alpha ^{\prime}+\beta ^{\prime}$|⁠. Suppose that |$E_{\alpha },F_{\alpha },E_{\beta },F_{\beta }$| have been already constructed. Then we define \begin{equation*} E_{\gamma}:=E_{\alpha} E_{\beta}-v^{(\alpha,\beta)}E_{\beta} E_{\alpha},\ F_{\gamma}:=F_{\beta} F_{\alpha}-v^{-(\alpha,\beta)}F_{\alpha} F_{\beta}. \end{equation*} According to [13], we have |$[E_{\gamma },F_{\gamma }]=a(\gamma )\frac{K_{\gamma }-K_{\gamma }^{-1}}{v_{\gamma }-v_{\gamma }^{-1}}$| for certain constants |$a(\gamma )\in{\mathbb{C}}(v^{1/\mathsf{N}})$|⁠, where |$v_{\gamma }:=v^{(\gamma ,\gamma )/2}$| (note that |$a(\alpha _i)=1$| and |$v_{\alpha _i}=v_i$|⁠). For |$\gamma \in \Delta _+$|⁠, define \begin{equation*} R_{\gamma}:=\textrm{exp}_{v_{\gamma}^{-1}} \left(\frac{v_{\gamma}-v_{\gamma}^{-1}}{a(\gamma)}E_{\gamma}\otimes F_{\gamma}\right), \end{equation*} where |$\textrm{exp}_v(x):=\sum _{r=0}^{\infty } \frac{x^r}{(r)_v!}, (r)_v!:=(1)_v\cdots (r)_v, (r)_v:=\frac{1-v^r}{1-v}$|⁠. The following is due to [13]. Theorem 3.6. ([13]). Fix a convex ordering |$\prec $| on |$\Delta _+$|⁠. Then |$\Theta ^{\textrm{op}}=\prod _{\gamma \in \Delta _+}R_{\gamma }$|⁠, where the order in the product coincides with the ordering |$\prec $|⁠. The explicit computations of |$\textbf{D}_1$| below are based on the special choice of convex orderings. We choose two convex orderings |$\prec _{\pm }$| on |$\Delta _+$| in such a way that |$\epsilon ^{\pm }_{ij}=-1\Rightarrow \alpha _i\prec _{\pm } \alpha _j$| (as shown in [23], any ordering on simple positive roots can be extended to a convex ordering on |$\Delta _+$|⁠). This choice is motivated by Proposition 3.7 below. To state the result, define \begin{equation} C^{\prime}_V=\textrm{tr}_V(\textrm{id}\otimes \pi_V)\left(\prod R_{\alpha_i}^{\textrm{op}}\cdot (R^0)^{\textrm{op}}\cdot \prod R_{\alpha_i}\cdot R^0\cdot (1\otimes K_{2\rho})\right), \end{equation} (3.1) where the 1st and the 2nd products are over all simple positive roots ordered according to |$\prec _-$| and |$\prec _+$|⁠, respectively, whereas |$(\textrm{id}\otimes \pi _V)(R^0)=(\textrm{id}\otimes \pi _V)((R^0)^{\textrm{op}})$| are understood as before. We define |$\bar{\textbf{D}}_{V}:=\tilde{\textbf{D}}_{C^{\prime}_V}$|⁠. Proposition 3.7. We have |$\tilde{\textbf{D}}_V=\bar{\textbf{D}}_V$|⁠. Proof. For |$\gamma =\sum _{i=1}^n m_i\alpha _i\in \Delta _+\ (m_i\in{\mathbb{Z}}_{\geq 0})$|⁠, define |$e_{\gamma },f_{\gamma }\in U_v(\mathfrak{g})$| via |$e_{\gamma }:=E_{\gamma }\cdot \prod _{i,k=1}^n L_k^{m_i\mathsf{n}^+_{ik}}$| and |$f_{\gamma }:=\prod _{i,k=1}^n L_k^{-m_i\mathsf{n}^-_{ik}}\cdot F_{\gamma }$|⁠, so that |$e_{\alpha _i}=e_i, f_{\alpha _i}=f_i$| as defined in Section 2.2. The proof of Proposition 3.7 is based on the following properties of these elements |$\{e_{\gamma }, f_{\gamma }\}_{\gamma \in \Delta _+}$| established in [22, Propositions 2.2.4 and 2.2.5]. Lemma 3.8. (a) For |$\gamma \in \Delta _+$|⁠, we have |$e_{\gamma }\in U^+_{\mathsf{n}^+}(\mathfrak{g})$| and |$f_{\gamma }\in U^-_{\mathsf{n}^-}(\mathfrak{g})$|⁠. (b) If |$\gamma \in \Delta _+$| is not a simple root, then |$\chi ^+(e_{\gamma })=0$| and |$\chi ^-(f_{\gamma })=0$|⁠. We recall the proof of this Lemma to make our exposition self-contained. Proof. (a) The proof is by induction used in the above definition of the root vectors |$E_{\gamma },F_{\gamma }$|⁠. The claim is trivial when |$\gamma $| is a simple root. For the remaining cases, let |$\alpha ,\beta ,\gamma \in \Delta _+$| be as above and assume that we have already established the inclusions |$e_{\alpha },e_{\beta }\in U^+_{\mathsf{n}^+}(\mathfrak{g})$| and |$f_{\alpha },f_{\beta }\in U^-_{\mathsf{n}^-}(\mathfrak{g})$|⁠. Let us write |$\alpha =\sum _{i=1}^n m_i\alpha _i, \beta =\sum _{i=1}^n m^{\prime}_i\alpha _i$|⁠. Then \begin{align*} & E_{\gamma}=\left(v^{-\sum_{i,k=1}^n m_i\mathsf{n}^+_{ik}(\omega_k,\beta)}e_{\alpha} e_{\beta}- v^{(\alpha,\beta)-\sum_{i,k=1}^n m^{\prime}_i\mathsf{n}^+_{ik}(\omega_k,\alpha)}e_{\beta} e_{\alpha}\right) \cdot \prod_{i,k=1}^n L_k^{-(m_i+m^{\prime}_i)\mathsf{n}^+_{ik}},\\ & F_{\gamma}=\prod_{i,k=1}^n L_k^{(m_i+m^{\prime}_i)\mathsf{n}^-_{ik}}\cdot \left(v^{\sum_{i,k=1}^n m_i\mathsf{n}^-_{ik}(\omega_k,\beta)}f_{\beta} f_{\alpha}-v^{-(\alpha,\beta)+ \sum_{i,k=1}^n m^{\prime}_i\mathsf{n}^-_{ik}(\omega_k,\alpha)}f_{\alpha} f_{\beta}\right), \end{align*} so that \begin{equation} e_{\gamma}= v^{-\sum_{i,k=1}^n m_i\mathsf{n}^+_{ik}(\omega_k,\beta)}e_{\alpha} e_{\beta}-v^{(\alpha,\beta)- \sum_{i,k=1}^n m^{\prime}_i\mathsf{n}^+_{ik}(\omega_k,\alpha)}e_{\beta} e_{\alpha} \end{equation} (3.2) and \begin{equation} f_{\gamma}= v^{\sum_{i,k=1}^n m_i\mathsf{n}^-_{ik}(\omega_k,\beta)}f_{\beta} f_{\alpha}- v^{-(\alpha,\beta)+\sum_{i,k=1}^n m^{\prime}_i\mathsf{n}^-_{ik}(\omega_k,\alpha)}f_{\alpha} f_{\beta}. \end{equation} (3.3) Thus, |$e_{\gamma }\in U^+_{\mathsf{n}^+}(\mathfrak{g})$| and |$f_{\gamma }\in U^-_{\mathsf{n}^-}(\mathfrak{g})$|⁠, which completes our inductive step. Part (a) follows. (b) Due to the formulas (3.2, 3.3), it suffices to prove |$\chi ^+(e_{\gamma })=0$| and |$\chi ^-(f_{\gamma })=0$| for |$\gamma =\alpha +\beta $| with |$\alpha =\alpha _i, \beta =\alpha _j$|⁠. In the former case, we get \begin{equation*} e_{\gamma}= v^{-{\mathsf{d}}_j\mathsf{n}^+_{ij}}e_ie_j-v^{b_{ij}-{\mathsf{d}}_i\mathsf{n}^+_{ji}}e_je_i= v^{-{\mathsf{d}}_j\mathsf{n}^+_{ij}}[e_i,e_j], \end{equation*} since |${\mathsf{d}}_j\mathsf{n}^+_{ij}-{\mathsf{d}}_i\mathsf{n}^+_{ji}=\epsilon ^+_{ij}b_{ij}=-b_{ij}$| as |$\alpha _i\prec _+ \alpha _j$|⁠. Hence, |$\chi ^+(e_{\gamma })=v^{-{\mathsf{d}}_j\mathsf{n}^+_{ij}}[\chi ^+(e_i),\chi ^+(e_j)]=0$|⁠. In the latter case, we get \begin{equation*} f_{\gamma}= v^{{\mathsf{d}}_j\mathsf{n}^-_{ij}}f_jf_i-v^{-b_{ij}+{\mathsf{d}}_i\mathsf{n}^-_{ji}}f_if_j= v^{{\mathsf{d}}_j\mathsf{n}^-_{ij}}[f_j,f_i], \end{equation*} since |${\mathsf{d}}_j\mathsf{n}^-_{ij}-{\mathsf{d}}_i\mathsf{n}^-_{ji}=\epsilon ^-_{ij}b_{ij}=-b_{ij}$| as |$\alpha _i\prec _- \alpha _j$|⁠. Thus, |$\chi ^-(f_{\gamma })=v^{{\mathsf{d}}_j\mathsf{n}^-_{ij}}[\chi ^-(f_j),\chi ^-(f_i)]=0$|⁠. Tracing back the definition of |$\tilde{\textbf{D}}_V$|⁠, Lemma 3.8 implies that |$R_{\gamma },R_{\gamma }^{\textrm{op}}$| give trivial contributions to |$\tilde{\textbf{D}}_V$| unless |$\gamma \in \Delta _+$| is a simple root, cf. [6, Lemma 5.2] and [7, Proposition 3.6]. Hence, the equality |$\tilde{\textbf{D}}_V=\bar{\textbf{D}}_V$|⁠. 3.9 Explicit formulas and classification in type |$A_{n-1}$| Recall explicit formulas for the action of |$U_v(\mathfrak{s}\mathfrak{l}_n)$| on its 1st fundamental representation |$V_1$|⁠. The space |$V_1$| has a basis |$\{w_1,\ldots ,w_n\}$|⁠, in which the action is given by the following formulas: \begin{equation*} E_i(w_j)=\delta_{j,i+1}w_{j-1},\ F_i(w_j)=\delta_{j,i}w_{j+1},\ L_i(w_j)=v^{-\frac{i}{n}+\delta_{j\leq i}}w_j,\ K_i(w_j)=v^{\delta_{j,i}-\delta_{j,i+1}}w_j\end{equation*} for any |$1\leq i<n, 1\leq j\leq n$|⁠. Let |$\varpi _1,\ldots ,\varpi _n$| be the weights of |$w_1,\ldots ,w_n$|⁠, respectively, so that |$(\varpi _i,\varpi _j)=\delta _{i,j}-1/n$|⁠. Recall that the simple roots are given by |$\alpha _i=\varpi _i-\varpi _{i+1}\ (1\leq i\leq n-1)$|⁠, while |$\rho =\sum _{j=1}^n \frac{n+1-2j}{2}\varpi _j$|⁠. According to Proposition 3.7, to compute |$\textbf{D}_1$| explicitly, we should evaluate |$C^{\prime}_{V_1}$|⁠, replace |$E_i,F_i$| by |$e_i,f_i$| and |$L_p$|⁠, moving the latter to the middle part, apply |$\chi ^{\pm }$| as in [6] to obtain the difference operator |$\tilde{\textbf{D}}_1=\bar{\textbf{D}}_{V_1}$|⁠, conjugate by |$e^{\rho }$|⁠. Note that the operators |$\{E_i^r,F_i^r\}^{r>1}_{1\leq i\leq n-1}$| act trivially on |$V_1$|⁠. Hence, applying formula (3.1), we can replace |$R_{\alpha _i}$| by |$\bar{R}_{\alpha _i}:=1+(v-v^{-1})E_i\otimes F_i$|⁠. Let us now compute all the nonzero terms contributing to |$C^{\prime}_{V_1}$|⁠: Picking |$1$| out of each |$\bar{R}^{\textrm{op}}_{\alpha _i}, \bar{R}_{\alpha _i}$|⁠, we recover |$\sum _{j=1}^n v^{n+1-2j}\cdot K_{2\varpi _j}$|⁠. Picking nontrivial terms only at |$\bar{R}_{\alpha _j}^{\textrm{op}},\bar{R}_{\alpha _i}$|⁠, the result does not depend on |$\textrm{Or}^{\pm }$| (hence, the orderings |$\prec _{\pm }$|⁠) and the total contribution is |$\sum _{i=1}^{n-1} (v-v^{-1})^2 v^{n+1-2i}\cdot F_iK_{\varpi _{i+1}}E_i K_{\varpi _i}$|⁠. Rewriting in terms of |$e_i,f_i$| and |$L_p$|⁠, we get |$(v-v^{-1})^2\sum _{i=1}^{n-1} v^{n-2i+(\mathsf{n}^+_{ii}-\mathsf{n}^-_{ii})}\cdot f_i K_{\varpi _i+\varpi _{i+1}}\prod _{p=1}^{n-1} L_p^{\mathsf{n}^-_{ip}-\mathsf{n}^+_{ip}} e_i$|⁠. The computation of the remaining terms is based on the following obvious formulas \begin{align*} & F_{i_k}\cdots F_{i_2}F_{i_1}(w_i)=\delta_{i_1,i}\delta_{i_2,i_1+1}\cdots\delta_{i_k,i_{k-1}+1}w_{i+k},\\ & E_{j_k}\cdots E_{j_2}E_{j_1}(w_j)=\delta_{j_1,j-1}\delta_{j_2,j_1-1}\cdots\delta_{j_k,j_{k-1}-1}w_{j-k}. \end{align*} Hence, picking nontrivial terms only at |$\bar{R}^{\textrm{op}}_{\alpha _{j_1}},\ldots , \bar{R}^{\textrm{op}}_{\alpha _{j_{k^{\prime}}}}, \bar{R}_{\alpha _{i_k}},\ldots ,\bar{R}_{\alpha _{i_1}}$| (in the order listed) is possible only if |$i_k\prec _+ \cdots \prec _+ i_1,\ j_1\prec _- \cdots \prec _- j_{k^{\prime}}$|⁠, and gives a nonzero contribution to |$C^{\prime}_{V_1}$| if and only if |$k=k^{\prime},\ i_k=i_{k-1}+1=\ldots =i_1+k-1$|⁠, and |$i_a=j_a$| for |$1\leq a\leq k$|⁠. Thus, the remaining terms contributing to |$C^{\prime}_{V_1}$| depend on |$\prec _{\pm }$| (only on |$\textrm{Or}^{\pm }$|⁠) and give in total \begin{equation*} \sum_{1\leq i<j-1\leq n-1}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-2,j-1}=\pm 1} (v-v^{-1})^{2(j-i)}v^{n+1-2i}\cdot F_i\cdots F_{j-1}K_{\varpi_j}E_{j-1}\cdots E_i K_{\varpi_i}. \end{equation*} Rewriting this in terms of |$e_i,f_i$| and |$L_p$|⁠, and moving the latter to the middle, we get \begin{equation*} \sum (v-v^{-1})^{2(j-i)}v^{n-2i+\sum_{i\leq a\leq b\leq j-1}(\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})}\cdot f_i\cdots f_{j-1}\cdot K_{\varpi_i+\varpi_j}\prod_{p=1}^{n-1} L_p^{\sum_{s=i}^{j-1} (\mathsf{n}^-_{sp}-\mathsf{n}^+_{sp})}\cdot e_{j-1}\cdots e_i, \end{equation*} where the sum is over all |$1\leq i<j-1\leq n-1$| such that |$\epsilon ^{\pm }_{i,i+1}=\ldots =\epsilon ^{\pm }_{j-2,j-1}=\pm 1$|⁠. Note that |$L_p=K_{\varpi _1+\ldots +\varpi _p}$|⁠. Set |$\mathsf{m}_{ik}:=\sum _{p=k}^{n-1} (\mathsf{n}^-_{ip}-\mathsf{n}^+_{ip})$|⁠. Then the Cartan part above equals |$K_{\varpi _i+\varpi _j}\prod _{p=1}^{n-1} L_p^{\sum _{s=i}^{j-1} (\mathsf{n}^-_{sp}-\mathsf{n}^+_{sp})}= K_{\sum _{k=1}^n (\sum _{s=i}^{j-1} \mathsf{m}_{sk}+\delta _{k,i}+\delta _{k,j})\varpi _k}$|⁠. We have listed all the nonzero terms contributing to |$C^{\prime}_{V_1}$|⁠. To obtain the desired difference operator |$\tilde{\textbf{D}}_1$|⁠, apply the characters |$\chi ^{\pm }$| with |$\chi ^+(e_i)=c^+_i, \chi ^-(\,f_i)=c^-_i$| as in [6, Lemma 5.2]. Set |$b_i:=(v-v^{-1})^2v^{\mathsf{n}^+_{ii}-\mathsf{n}^-_{ii}}c^+_ic^-_i$|⁠. Then we have \begin{multline} \tilde{\textbf{D}}_1=\sum_{j=1}^n v^{n+1-2j} T_{2\varpi_j}+ \sum_{i=1}^{n-1} b_iv^{n-2i}\cdot e^{-\alpha_i}T_{\sum_{k=1}^n (\mathsf{m}_{ik}+\delta_{k,i}+\delta_{k,i+1})\varpi_k}+\\ \sum_{1\leq i<j-1\leq n-1}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-2,j-1}=\pm 1} b_i\cdots b_{j-1} v^{n-2i+\sum_{i\leq a<b\leq j-1} (\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})}\times\\ e^{-\sum_{s=i}^{j-1} \alpha_s}T_{\sum_{k=1}^n (\sum_{s=i}^{j-1} \mathsf{m}_{sk}+\delta_{k,i}+\delta_{k,j})\varpi_k}. \end{multline} (3.4) Conjugating this by |$e^{\rho }$|⁠, we finally obtain the explicit formula for the 1st hamiltonian |$\textbf{D}_1$| of the type |$A_{n-1}$| modified quantum difference Toda system: \begin{multline} \textbf{D}_1=\sum_{j=1}^n T_{2\varpi_j}+ \sum_{i=1}^{n-1} b_iv^{\sum_{k=1}^n \frac{2k-n-1}{2}\mathsf{m}_{ik}}\cdot e^{-\alpha_i}T_{\sum_{k=1}^n (\mathsf{m}_{ik}+\delta_{k,i}+\delta_{k,i+1})\varpi_k}+\\ \sum_{1\leq i<j-1\leq n-1}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-2,j-1}=\pm 1} b_i\cdots b_{j-1} v^{j-i-1+\sum_{i\leq a<b\leq j-1} (\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})+\sum_{k=1}^n\sum_{s=i}^{j-1}\frac{2k-n-1}{2}\mathsf{m}_{sk}}\times\\ e^{-\sum_{s=i}^{j-1} \alpha_s}T_{\sum_{k=1}^n (\sum_{s=i}^{j-1} \mathsf{m}_{sk}+\delta_{k,i}+\delta_{k,j})\varpi_k}. \end{multline} (3.5) Remark 3.10. If |$\epsilon ^+=\epsilon ^-$|⁠, then the last sum is vacuous. If we also set |$\mathsf{n}^+=\mathsf{n}^-$| and |$c^{\pm }_i=\pm 1$| for all |$i$|⁠, then we recover the formula [6, (5.7)] for the 1st hamiltonian of the type |$A_{n-1}$| quantum difference Toda lattice: \begin{equation} \textbf{D}_1=\sum_{j=1}^n T_{2\varpi_j}-(v-v^{-1})^2\sum_{i=1}^{n-1} e^{-\alpha_i}T_{\varpi_i+\varpi_{i+1}}. \end{equation} (3.6) Let |${\mathcal{A}}_n$| be the associative |${\mathbb{C}}(v^{1/\mathsf{N}})$|-algebra generated by |$\{{\mathsf{w}}^{\pm 1}_j, {\mathsf{D}}^{\pm 1}_j\}_{j=1}^n$| with the defining relations \begin{equation} [{\mathsf{w}}_i,{\mathsf{w}}_j]=[{\mathsf{D}}_i,{\mathsf{D}}_j]=0,\ {\mathsf{w}}^{\pm 1}_i{\mathsf{w}}^{\mp 1}_i={\mathsf{D}}^{\pm 1}_i{\mathsf{D}}^{\mp 1}_i=1,\ {\mathsf{D}}_i{\mathsf{w}}_j=v^{\delta_{i,j}}{\mathsf{w}}_j{\mathsf{D}}_i. \end{equation} (3.7) Define |$\bar{{\mathcal{A}}}_n$| as the quotient of the |${\mathbb{C}}(v^{1/\mathsf{N}})$|-subalgebra generated by |$\{{\mathsf{w}}_j^{\pm 1}, ({\mathsf{D}}_i/{\mathsf{D}}_{i+1})^{\pm 1}\}_{1\leq i<n}^{1\leq j\leq n}$| by the relation |${\mathsf{w}}_1\cdots{\mathsf{w}}_n=1$|⁠. Consider the anti-isomorphism from the algebra |$\bar{{\mathcal{A}}}_n$| to the algebra |${\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{l}_n})$| of Section 2.5, sending |${\mathsf{w}}_j\mapsto T_{-\varpi _j}, {\mathsf{D}}_i/{\mathsf{D}}_{i+1}\mapsto e^{-\alpha _i}$|⁠. Then the hamiltonian |$\textbf{D}_1$| is the image of the following element |$\mathsf{H}=\mathsf{H}(\epsilon ^{\pm }, \mathsf{n}^{\pm }, c^{\pm })$| of |$\bar{{\mathcal{A}}}_n$|⁠: \begin{multline} \mathsf{H}(\epsilon^{\pm}, \mathsf{n}^{\pm}, c^{\pm})= \sum_{j=1}^n {\mathsf{w}}_j^{-2} + \sum_{i=1}^{n-1} b_i v^{\sum_{k=1}^n \frac{2k-n-1}{2}\mathsf{m}_{ik}}\cdot \prod_{k=1}^n {\mathsf{w}}_k^{-\mathsf{m}_{ik}-\delta_{k,i}-\delta_{k,i+1}}\cdot \frac{{\mathsf{D}}_i}{{\mathsf{D}}_{i+1}}+\\ \sum_{1\leq i<j-1\leq n-1}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-2,j-1}=\pm 1} b_i\cdots b_{j-1} v^{j-i-1+\sum_{i\leq a<b\leq j-1} (\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})+ \sum_{k=1}^n\sum_{s=i}^{j-1}\frac{2k-n-1}{2}\mathsf{m}_{sk}}\times\\ \prod_{k=1}^n {\mathsf{w}}_k^{-\sum_{s=i}^{j-1} \mathsf{m}_{sk}-\delta_{k,i}-\delta_{k,j}}\cdot \frac{{\mathsf{D}}_i}{{\mathsf{D}}_j}. \end{multline} (3.8) The following is the key property of |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| in type |$A$|⁠. Proposition 3.11. |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| depends only on |$\vec{\epsilon }=(\epsilon _{n-2},\ldots ,\epsilon _1)\in \{-1,0,1\}^{n-2}$| with |$\epsilon _i:=\frac{\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}}{2}$|⁠, up to algebra automorphisms of |$\bar{{\mathcal{A}}}_n$|⁠. This result implies that given two pairs of Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| and |$(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| with |$\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}=\tilde{\epsilon }^+_{i,i+1}-\tilde{\epsilon }^-_{i,i+1}$|⁠, there exists an algebra automorphism of |${\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{l}_n})$| that maps the 1st hamiltonian |$\textbf{D}_1(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |$\textbf{D}_1(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. As we will see in Appendix B, the same automorphism maps the modified quantum Toda system |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |${\mathcal{T}}(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. We present the proof of Proposition 3.11 in Appendix A. 3.12 Explicit formulas and classification in type |$C_n$| Recall explicit formulas for the action of |$U_v(\mathfrak{s}\mathfrak{p}_{2n})$| on its 1st fundamental representation |$V_1$|⁠. The space |$V_1$| has a basis |$\{w_1,\ldots ,w_n,w_{n+1},\ldots ,w_{2n}\}$|⁠, in which the action is given via \begin{align*} & E_i(w_j)=\delta_{j,i+1}w_{j-1},\ E_i(w_{n+j})=\delta_{j,i}w_{n+j+1},\ E_n(w_j)=0,\ E_n(w_{n+j})=\delta_{j,n}w_n,\\ & F_i(w_j)=\delta_{j,i}w_{j+1},\ F_i(w_{n+j})=\delta_{j,i+1}w_{n+j-1},\ F_n(w_j)=\delta_{j,n}w_{2n},\ F_n(w_{n+j})=0,\\ & L_i(w_j)=v^{\delta_{j\leq i}}w_j,\ L_i(w_{n+j})=v^{-\delta_{j\leq i}} w_j,\ L_n(w_j)=vw_j,\ L_n(w_{n+j})=v^{-1}w_{n+j},\\ & K_i(w_j)=v^{\delta_{j,i}-\delta_{j,i+1}}w_j,\ K_i(w_{n+j})=v^{-\delta_{j,i}+\delta_{j,i+1}}w_{n+j},\\ & K_n(w_j)=v^{2\delta_{j,n}}w_j,\ K_n(w_{n+j})=v^{-2\delta_{j,n}}w_{n+j} \end{align*} for any |$1\leq i<n, 1\leq j\leq n$|⁠. Let |$\varpi _j$| be the weight of |$w_j\ (1\leq j\leq n)$|⁠, so that the weight of |$w_{n+j}$| equals |$-\varpi _j$|⁠, while |$(\varpi _i,\varpi _j)=\delta _{i,j}$|⁠. Recall that the simple positive roots are given by |$\alpha _i=\varpi _i-\varpi _{i+1}\ (1\leq i\leq n-1)$| and |$\alpha _n=2\varpi _n$|⁠, while |$\rho =\sum _{i=1}^n (n+1-i)\varpi _i$| and |${\mathsf{d}}_1=\ldots ={\mathsf{d}}_{n-1}=1, {\mathsf{d}}_n=2$|⁠. To compute |$\textbf{D}_1$| explicitly, we use the same strategy as in type |$A$|⁠. Note that the operators |$\{E_i^r,F_i^r\}^{r>1}_{1\leq i\leq n}$| act trivially on |$V_1$|⁠. Therefore, applying formula (3.1), we can replace |$R_{\alpha _i}$| by |$\bar{R}_{\alpha _i}:=1+(v_i-v_i^{-1})E_i\otimes F_i$|⁠. Let us now compute all the nonzero terms contributing to |$C^{\prime}_{V_1}$|⁠: Picking |$1$| out of each |$\bar{R}^{\textrm{op}}_{\alpha _i}, \bar{R}_{\alpha _i}$|⁠, we recover |$\sum _{i=1}^n \left (v^{2(n+1-i)}\cdot K_{2\varpi _i}+v^{-2(n+1-i)}\cdot\right.$||$\left.K_{-2\varpi _i}\right )$| Picking nontrivial terms only at |$\bar{R}_{\alpha _j}^{\textrm{op}}, \bar{R}_{\alpha _i}$|⁠, the result does not depend on |$\textrm{Or}^{\pm }$| (hence, the orderings |$\prec _{\pm }$|⁠) and the total contribution of the nonzero terms equals \begin{align*} & \sum_{i=1}^{n-1} (v-v^{-1})^2\left(v^{2(n+1-i)}F_iK_{\varpi_{i+1}}E_iK_{\varpi_i}+ v^{-2(n-i)}F_iK_{-\varpi_i}E_iK_{-\varpi_{i+1}}\right)+\\ & (v^2-v^{-2})^2v^2F_nK_{-\varpi_n}E_nK_{\varpi_n}. \end{align*} The other terms contributing to |$C^{\prime}_{V_1}$| depend on |$\prec _{\pm }$| (only on |$\textrm{Or}^{\pm }$|⁠). Picking nontrivial terms only at |$\bar{R}^{\textrm{op}}_{\alpha _{j_1}},\ldots , \bar{R}^{\textrm{op}}_{\alpha _{j_{k^{\prime}}}}, \bar{R}_{\alpha _{i_k}},\ldots ,\bar{R}_{\alpha _{i_1}}$| (in the order listed) is possible only if |$i_k\prec _+ \cdots \prec _+ i_1,\ j_1\prec _- \cdots \prec _- j_{k^{\prime}}$|⁠, and gives a nonzero contribution to |$C^{\prime}_{V_1}$| if and only if |$k=k^{\prime}, i_k=i_{k-1}\pm 1=\ldots =i_1\pm (k-1)$| (the sign stays the same everywhere), and |$i_a=j_a$| for |$1\leq a\leq k$|⁠. When computing these contributions, we shall distinguish between the two cases: |$\max (i_1,i_k)=n$| and |$\max (i_1,i_k)<n$|⁠. The total contribution of such terms with |$k>1$| equals \begin{align*} & \sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\pm 1} (v-v^{-1})^{2(j-i+1)}v^{2(n+1-i)}\cdot F_i\cdots F_{j}K_{\varpi_{j+1}}E_{j}\cdots E_i K_{\varpi_i}+\\ & \sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\pm 1} (v-v^{-1})^{2(n-i)}(v^2-v^{-2})^{2}v^{2(n+1-i)}\cdot F_i\cdots F_{n}K_{-\varpi_n}E_{n}\cdots E_i K_{\varpi_i}+\\ & \sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\mp 1} (v-v^{-1})^{2(j-i+1)}v^{-2(n-j)}\cdot F_j\cdots F_{i}K_{-\varpi_{i}}E_{i}\cdots E_j K_{-\varpi_{j+1}}+\\ & \sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\mp 1} (v-v^{-1})^{2(n-i)}(v^2-v^{-2})^{2}v^{2}\cdot F_n\cdots F_{i}K_{-\varpi_i}E_{i}\cdots E_n K_{\varpi_n}. \end{align*} We have listed all the nonzero terms contributing to |$C^{\prime}_{V_1}$|⁠. To obtain |$\tilde{\textbf{D}}_1=\bar{\textbf{D}}_{V_1}$|⁠, we should rewrite the above formulas via |$e_i,f_i$| and |$L_p=K_{\varpi _1+\ldots +\varpi _p}\ (1\leq p\leq n)$|⁠, moving all the Cartan terms to the middle, and then apply the characters |$\chi ^{\pm }$| with |$\chi ^+(e_i)=c^+_i, \chi ^-(f_i)=c^-_i$|⁠. Conjugating further by |$e^{\rho }$|⁠, we obtain the explicit formula for the 1st hamiltonian |$\textbf{D}_1$| of the type |$C_n$| modified quantum difference Toda system. To write it down, define constants |$b_i,\mathsf{m}_{ik}$| via |$b_i:=(v_i-v_i^{-1})^2v_i^{\mathsf{n}^+_{ii}-\mathsf{n}^-_{ii}}c^+_ic^-_i$| and |$\mathsf{m}_{ik}:=\sum _{p=k}^n(\mathsf{n}^-_{ip}-\mathsf{n}^+_{ip})$|⁠. Then we have \begin{multline} \textbf{D}_1= \sum_{i=1}^n (T_{2\varpi_i}+T_{-2\varpi_i})+ b_n v^{\sum_{k=1}^n (k-n-1) \mathsf{m}_{nk}}\cdot e^{-\alpha_n}T_{\sum_{k=1}^n \mathsf{m}_{nk}\varpi_k}+\\ \sum_{i=1}^{n-1} b_iv^{\sum_{k=1}^n (k-n-1) \mathsf{m}_{ik}}\cdot e^{-\alpha_i}\left(T_{\sum_{k=1}^n (\mathsf{m}_{ik}+\delta_{k,i}+\delta_{k,i+1})\varpi_k}+T_{\sum_{k=1}^n (\mathsf{m}_{ik}-\delta_{k,i}-\delta_{k,i+1})\varpi_k}\right)+\\ \sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\pm 1} b_i\cdots b_j v^{j-i+\sum_{i\leq a<b\leq j} (\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})+\sum_{k=1}^n\sum_{s=i}^j (k-n-1) \mathsf{m}_{sk}}\times\\ e^{-\alpha_i-\ldots-\alpha_j}T_{\sum_{k=1}^n(\sum_{s=i}^j \mathsf{m}_{sk}+\delta_{k,i}+\delta_{k,j+1})\varpi_k}+\\ \sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\pm 1} b_i\cdots b_n v^{n+1-i+\sum_{i\leq a<b\leq n}(\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})(1+\delta_{b,n})+\sum_{k=1}^n\sum_{s=i}^n (k-n-1) \mathsf{m}_{sk}}\times\\ e^{-\alpha_i-\ldots-\alpha_n}T_{\sum_{k=1}^n(\sum_{s=i}^n \mathsf{m}_{sk}+\delta_{k,i}-\delta_{k,n})\varpi_k}+\\ \sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\mp 1} b_i\cdots b_j v^{j-i+\sum_{i\leq a<b\leq j}(\mathsf{n}^+_{ba}-\mathsf{n}^-_{ba})+\sum_{k=1}^n\sum_{s=i}^j (k-n-1) \mathsf{m}_{sk}}\times\\ e^{-\alpha_i-\ldots-\alpha_j}T_{\sum_{k=1}^n(\sum_{s=i}^j \mathsf{m}_{sk}-\delta_{k,i}-\delta_{k,j+1})\varpi_k}+\\ \sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\mp 1} b_i\cdots b_n v^{n+1-i+\sum_{i\leq a<b\leq n} (\mathsf{n}^+_{ba}-\mathsf{n}^-_{ba})+\sum_{k=1}^n\sum_{s=i}^n (k-n-1) \mathsf{m}_{sk}}\times\\ e^{-\alpha_i-\ldots-\alpha_n}T_{\sum_{k=1}^n(\sum_{s=i}^n \mathsf{m}_{sk}-\delta_{k,i}+\delta_{k,n})\varpi_k}. \end{multline} (3.9) Remark 3.13. If |$\epsilon ^+=\epsilon ^-$|⁠, then the last four sums are vacuous. If we also set |$\mathsf{n}^+=\mathsf{n}^-$| and |$c^{\pm }_i=\pm 1$| for all |$i$|⁠, then we obtain the formula for the 1st hamiltonian of the type |$C_n$| quantum difference Toda lattice as defined in [6] (we write down this formula as we could not find it in the literature, even though it can be derived completely analogously to [6, (5.7)], cf. [7, the end of Section 3]): \begin{equation} \textbf{D}_1= \sum_{i=1}^n (T_{2\varpi_i}+T_{-2\varpi_i})- (v-v^{-1})^2\sum_{i=1}^{n-1} e^{-\alpha_i}\left(T_{\varpi_i+\varpi_{i+1}}+T_{-\varpi_i-\varpi_{i+1}}\right)- (v^2-v^{-2})^2 e^{-\alpha_n}. \end{equation} (3.10) Let |${\mathcal C}_n$| be the |${\mathbb{C}}(v^{1/\mathsf{N}})$|-subalgebra of |${\mathcal{A}}_n$| generated by |$\{{\mathsf{w}}^{\pm 1}_j, ({\mathsf{D}}_i/{\mathsf{D}}_{i+1})^{\pm 1}, {\mathsf{D}}_n^{\pm 2}\}_{1\leq i<n}^{1\leq j\leq n}$| (note that |${\mathcal C}_n$| can be abstractly defined as the associative algebra generated by |$\{\tilde{{\mathsf{w}}}_i^{\pm 1},\tilde{{\mathsf{D}}}_i^{\pm 1}\}_{i=1}^n$| with the defining relations |$[\tilde{{\mathsf{w}}}_i,\tilde{{\mathsf{w}}}_j]=[\tilde{{\mathsf{D}}}_i,\tilde{{\mathsf{D}}}_j]=0, \tilde{{\mathsf{w}}}^{\pm 1}_i\tilde{{\mathsf{w}}}^{\mp 1}_i=\tilde{{\mathsf{D}}}^{\pm 1}_i\tilde{{\mathsf{D}}}^{\mp 1}_i=1, \tilde{{\mathsf{D}}}_i\tilde{{\mathsf{w}}}_j=v^{\delta _{i,j}{\mathsf{d}}_i}\tilde{{\mathsf{w}}}_j\tilde{{\mathsf{D}}}_i,$| where |${\mathsf{d}}_i=1+\delta _{i,n}$|⁠). Consider the anti-isomorphism from |${\mathcal C}_n$| to the algebra |${\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{p}_{2n}})$| of Section 2.5, sending |${\mathsf{w}}_j\mapsto T_{-\varpi _j}, {\mathsf{D}}_i/{\mathsf{D}}_{i+1}\mapsto e^{-\alpha _i}, {\mathsf{D}}_n^2\mapsto e^{-\alpha _n}$|⁠. Then the hamiltonian |$\textbf{D}_1$| is the image of the following element |$\mathsf{H}=\mathsf{H}(\epsilon ^{\pm }, \mathsf{n}^{\pm }, c^{\pm })$| of |${\mathcal C}_n$|⁠: \begin{multline} \mathsf{H}(\epsilon^{\pm}, \mathsf{n}^{\pm}, c^{\pm})= \sum_{i=1}^n ({\mathsf{w}}_i^{-2}+{\mathsf{w}}_i^2)+ b_n v^{\sum_{k=1}^n (k-n-1) \mathsf{m}_{nk}} \cdot \prod_{k=1}^n {\mathsf{w}}_k^{-\mathsf{m}_{nk}}\cdot{\mathsf{D}}_n^2+\\[-5pt] \sum_{i=1}^{n-1} b_iv^{\sum_{k=1}^n (k-n-1) \mathsf{m}_{ik}}\cdot \left(\prod_{k=1}^n {\mathsf{w}}_k^{-\mathsf{m}_{ik}-\delta_{k,i}-\delta_{k,i+1}}+ \prod_{k=1}^n {\mathsf{w}}_k^{-\mathsf{m}_{ik}+\delta_{k,i}+\delta_{k,i+1}}\right)\cdot\frac{{\mathsf{D}}_i}{{\mathsf{D}}_{i+1}}+\\ \sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\pm 1} b_i\cdots b_j v^{j-i+\sum_{i\leq a<b\leq j}(\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})+\sum_{k=1}^n\sum_{s=i}^j (k-n-1) \mathsf{m}_{sk}}\times\\ \prod_{k=1}^n {\mathsf{w}}_k^{-\sum_{s=i}^j \mathsf{m}_{sk}-\delta_{k,i}-\delta_{k,j+1}}\cdot \frac{{\mathsf{D}}_i}{{\mathsf{D}}_{j+1}}+\\ \sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\mp 1} b_i\cdots b_j v^{j-i+\sum_{i\leq a<b\leq j}(\mathsf{n}^+_{ba}-\mathsf{n}^-_{ba})+\sum_{k=1}^n\sum_{s=i}^j (k-n-1) \mathsf{m}_{sk}}\times\\ \prod_{k=1}^n {\mathsf{w}}_k^{-\sum_{s=i}^j \mathsf{m}_{sk}+\delta_{k,i}+\delta_{k,j+1}}\cdot\frac{{\mathsf{D}}_i}{{\mathsf{D}}_{j+1}}+\\ \sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\pm 1} b_i\cdots b_n v^{n+1-i+\sum_{i\leq a<b\leq n}(\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})(1+\delta_{b,n})+\sum_{k=1}^n\sum_{s=i}^n (k-n-1) \mathsf{m}_{sk}}\times\\ \prod_{k=1}^n {\mathsf{w}}_k^{-\sum_{s=i}^n \mathsf{m}_{sk}-\delta_{k,i}+\delta_{k,n}}\cdot{\mathsf{D}}_i{\mathsf{D}}_n+\\ \sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\mp 1} b_i\cdots b_n v^{n+1-i+\sum_{i\leq a<b\leq n} (\mathsf{n}^+_{ba}-\mathsf{n}^-_{ba})+\sum_{k=1}^n\sum_{s=i}^n (k-n-1) \mathsf{m}_{sk}}\times\\[-5pt] \prod_{k=1}^n {\mathsf{w}}_k^{-\sum_{s=i}^n \mathsf{m}_{sk}+\delta_{k,i}-\delta_{k,n}}\cdot{\mathsf{D}}_i{\mathsf{D}}_n. \end{multline} (3.11) The following is the key property of |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| in type |$C$|⁠. Proposition 3.14. |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| depends only on |$\vec{\epsilon }=(\epsilon _{n-1},\ldots ,\epsilon _1)\in \{-1,0,1\}^{n-1}$| with |$\epsilon _i:=\frac{\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}}{2}$|⁠, up to algebra automorphisms of |${\mathcal C}_n$|⁠. The proof of this result is completely analogous to that of Proposition 3.11 given in Appendix A, see also Remark A.1; we leave the details to the interested reader. Proposition 3.14 implies that given two pairs of Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| and |$(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| with |$\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}=\tilde{\epsilon }^+_{i,i+1}-\tilde{\epsilon }^-_{i,i+1}$|⁠, there exists an algebra automorphism of |${\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{p}_{2n}})$| that maps the 1st hamiltonian |$\textbf{D}_1(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |$\textbf{D}_1(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. As we will see in Appendix B, the same automorphism maps the modified quantum Toda system |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |${\mathcal{T}}(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. 3.15 Explicit formulas and classification in type |$D_n$| Recall explicit formulas for the action of |$U_v(\mathfrak{s}\mathfrak{o}_{2n})$| on its 1st fundamental representation |$V_1$|⁠. The space |$V_1$| has a basis |$\{w_1,\ldots ,w_{2n}\}$|⁠, in which the action is given via \begin{align*} & E_i(w_{2j-1})=\delta_{j,i+1}w_{2j-3},\ E_i(w_{2j})=\delta_{j,i}w_{2j+2},\\ & F_i(w_{2j-1})=\delta_{j,i}w_{2j+1},\ F_i(w_{2j})=\delta_{j,i+1}w_{2j-2},\\ & L_p(w_{2j-1})=v^{\delta_{j\leq p}}w_{2j-1}, \ L_p(w_{2j})=v^{-\delta_{j\leq p}}w_{2j},\\ & L_{n-1}(w_{2j-1})=v^{\frac{1}{2}-\delta_{j,n}}w_{2j-1}, \ L_{n-1}(w_{2j})=v^{-\frac{1}{2}+\delta_{j,n}}w_{2j},\\ & K_i(w_{2j-1})=v^{\delta_{j,i}-\delta_{j,i+1}}w_{2j-1},\ K_i(w_{2j})=v^{-\delta_{j,i}+\delta_{j,i+1}}w_{2j},\\ & E_n(w_{2j})=\delta_{j,n-1}w_{2n-1}+\delta_{j,n}w_{2n-3},\ F_n(w_{2j-1})=\delta_{j,n-1}w_{2n}+\delta_{j,n}w_{2n-2},\\ & E_n(w_{2j-1})=0,\ F_n(w_{2j})=0,\ L_n(w_{2j-1})=v^{\frac{1}{2}} w_{2j-1},\ L_n(w_{2j})=v^{-\frac{1}{2}}w_{2j},\\ & K_n(w_{2j-1})=v^{\delta_{j,n}+\delta_{j,n-1}}w_{2j-1},\ K_n(w_{2j})=v^{-\delta_{j,n}-\delta_{j,n-1}}w_{2j} \end{align*} for any |$1\leq p\leq n-2, 1\leq i<n, 1\leq j\leq n$|⁠. Let |$\varpi _j$| be the weight of |$w_{2j-1}\ (1\leq j\leq n)$|⁠, so that the weight of |$w_{2j}$| equals |$-\varpi _j$|⁠, while |$(\varpi _i,\varpi _j)=\delta _{i,j}$|⁠. Recall that the simple roots are given by |$\alpha _i=\varpi _i-\varpi _{i+1}\ (1\leq i\leq n-1)$| and |$\alpha _n=\varpi _{n-1}+\varpi _n$|⁠, while |$\rho =\sum _{i=1}^n (n-i)\varpi _i$| and |${\mathsf{d}}_1=\ldots ={\mathsf{d}}_{n}=1$|⁠. To compute |$\textbf{D}_1$| explicitly, we use the same strategy as in type |$A$|⁠. Similarly to the types |$A$| and |$C$| treated above, we note that the operators |$\{E_i^r,F_i^r\}^{r>1}_{1\leq i\leq n}$| act trivially on |$V_1$|⁠; hence, applying formula (3.1), we can replace |$R_{\alpha _i}$| by |$\bar{R}_{\alpha _i}:=1+(v-v^{-1})E_i\otimes F_i$|⁠. Let us now compute all the nonzero terms contributing to |$C^{\prime}_{V_1}$|⁠: Picking |$1$| out of each |$\bar{R}^{\textrm{op}}_{\alpha _i}, \bar{R}_{\alpha _i}$|⁠, we recover |$\sum _{i=1}^n \left (v^{2(n-i)}\cdot K_{2\varpi _i}+v^{-2(n-i)}\cdot\right.$||$\left. K_{-2\varpi _i}\right )$|⁠. Picking nontrivial terms only at |$\bar{R}_{\alpha _j}^{\textrm{op}}, \bar{R}_{\alpha _i}$|⁠, the result does not depend on |$\textrm{Or}^{\pm }$| (hence, the orderings |$\prec _{\pm }$|⁠) and the total contribution of the nonzero terms equals \begin{align*} & \sum_{i=1}^{n-1} (v-v^{-1})^2\left(v^{2(n-i)}F_iK_{\varpi_{i+1}}E_iK_{\varpi_i}+ v^{-2(n-i-1)}F_iK_{-\varpi_i}E_iK_{-\varpi_{i+1}}\right)+\\ & (v-v^{-1})^2\left(F_nK_{-\varpi_{n-1}}E_nK_{\varpi_n}+F_nK_{-\varpi_n}E_nK_{\varpi_{n-1}}\right). \end{align*} In contrast to the types |$A,C$| considered above, there is one more summand independent of the orientations. It arises by picking nontrivial terms only at |$\bar{R}^{\textrm{op}}_{\alpha _{n-1}},\bar{R}^{\textrm{op}}_{\alpha _n}$| and |$\bar{R}_{\alpha _{n-1}}, \bar{R}_{\alpha _n}$| (note that |$E_{n-1}E_{n}=E_{n}E_{n-1}, F_{n-1}F_{n}=F_{n}F_{n-1}$|⁠, due to the |$v$|-Serre relations) and equals \begin{equation*} (v-v^{-1})^4v^2F_nF_{n-1}K_{-\varpi_{n-1}}E_nE_{n-1}K_{\varpi_{n-1}}. \end{equation*} The contribution of the remaining terms to |$C^{\prime}_{V_1}$| depends on |$\textrm{Or}^{\pm }$|⁠. Tracing back explicit formulas for the action of |$U_v(\mathfrak{s}\mathfrak{o}_{2n})$| on |$V_1$|⁠, we see that the total sum of such terms equals \begin{align*} & \sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\pm 1} (v-v^{-1})^{2(j-i+1)}v^{2(n-i)}\cdot F_i\cdots F_{j}K_{\varpi_{j+1}}E_{j}\cdots E_i K_{\varpi_i}+\\ & \sum_{1\leq i<n-1}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-3,n-2}=\epsilon^{\pm}_{n-2,n}=\pm 1} (v-v^{-1})^{2(n-i)}v^{2(n-i)}\cdot \\ & F_i\cdots F_{n-2}F_nK_{-\varpi_n}E_{n}E_{n-2}\cdots E_i K_{\varpi_i}+\\[.5pt] & \sum_{1\leq i<n-1}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-2,n-1}=\epsilon^{\pm}_{n-2,n}=\pm 1} (v-v^{-1})^{2(n-i+1)}v^{2(n-i)}\cdot\\ & F_i\cdots F_{n-1}F_nK_{-\varpi_{n-1}}E_{n}E_{n-1}\cdots E_i K_{\varpi_i}+\\[.5pt] & \sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\mp 1} (v-v^{-1})^{2(j-i+1)}v^{-2(n-j-1)}\cdot F_j\cdots F_{i}K_{-\varpi_{i}}E_{i}\cdots E_j K_{-\varpi_{j+1}}+\\ & \sum_{1\leq i<n-1}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-3,n-2}=\epsilon^{\pm}_{n-2,n}=\mp 1} (v-v^{-1})^{2(n-i)}\cdot F_nF_{n-2}\cdots F_{i}K_{-\varpi_i}E_{i}\cdots E_{n-2}E_n K_{\varpi_n}+ \end{align*} \begin{align*} & \sum_{1\leq i<n-1}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-2,n-1}=\epsilon^{\pm}_{n-2,n}=\mp 1} (v-v^{-1})^{2(n-i+1)}v^2 \cdot \\ & F_nF_{n-1}\cdots F_i K_{-\varpi_i}E_{i}\cdots E_{n-1}E_n K_{\varpi_{n-1}}. \end{align*} We have listed all the nonzero terms contributing to |$C^{\prime}_{V_1}$|⁠. To obtain |$\tilde{\textbf{D}}_1=\bar{\textbf{D}}_{V_1}$|⁠, we should rewrite the above formulas via |$e_i,f_i$| and $$L_p=\begin{cases} K_{\varpi _1+\ldots +\varpi _p}, & \textrm{if}\ 1\leq p\leq n-2\\ K_{\frac{1}{2}(\varpi _1+\ldots +\varpi _{n-1}-\varpi _n)}, & \textrm{if}\ p=n-1\\ K_{\frac{1}{2}(\varpi _1+\ldots +\varpi _{n-1}+\varpi _n)}, & \textrm{if}\ p=n \end{cases},$$ moving all the Cartan terms to the middle, and then apply the characters |$\chi ^{\pm }$| with |$\chi ^+(e_i)=c^+_i, \chi ^-(f_i)=c^-_i$|⁠. Conjugating further by |$e^{\rho }$|⁠, we obtain the explicit formula for the 1st hamiltonian |$\textbf{D}_1$| of the type |$D_n$| modified quantum difference Toda system. To write it down, define $$\mathsf{m}_{ik}:=\begin{cases} \sum _{p=k}^{n-2}(\mathsf{n}^-_{ip}-\mathsf{n}^+_{ip})+\frac{1}{2}(\mathsf{n}^-_{i,n-1}-\mathsf{n}^+_{i,n-1})+\frac{1}{2}(\mathsf{n}^-_{in}-\mathsf{n}^+_{in}),\!\!\! & \textrm{if}\ k<n\\ -\frac{1}{2}(\mathsf{n}^-_{i,n-1}-\mathsf{n}^+_{i,n-1})+\frac{1}{2}(\mathsf{n}^-_{in}-\mathsf{n}^+_{in}), & \textrm{if}\ k=n \end{cases}$$ and |$b_i:=(v-v^{-1})^2v^{\mathsf{n}^+_{ii}-\mathsf{n}^-_{ii}}c^+_ic^-_i$|⁠. Then we have \begin{multline} \textbf{D}_1= \sum_{i=1}^n (T_{2\varpi_i}+T_{-2\varpi_i})+ b_n v^{\sum_{k=1}^n (k-n) \mathsf{m}_{nk}}\cdot e^{-\alpha_n}T_{\sum_{k=1}^n (\mathsf{m}_{nk}-\delta_{k,n-1}+\delta_{k,n})\varpi_k}+\\ b_n v^{-2+\sum_{k=1}^n (k-n) \mathsf{m}_{nk}}\cdot e^{-\alpha_n}T_{\sum_{k=1}^n (\mathsf{m}_{nk}+\delta_{k,n-1}-\delta_{k,n})\varpi_k}+\\ b_{n-1}b_nv^{(\mathsf{n}^+_{n-1,n}-\mathsf{n}^-_{n-1,n})+\sum_{k=1}^n (k-n) (\mathsf{m}_{n-1,k}+\mathsf{m}_{nk})}\cdot e^{-\alpha_{n-1}-\alpha_n}T_{\sum_{k=1}^n (\mathsf{m}_{n-1,k}+\mathsf{m}_{n,k})\varpi_k}+\\ \sum_{i=1}^{n-1} b_iv^{\sum_{k=1}^n (k-n) \mathsf{m}_{ik}}\cdot e^{-\alpha_i}\left(T_{\sum_{k=1}^n (\mathsf{m}_{ik}+\delta_{k,i}+\delta_{k,i+1})\varpi_k}+T_{\sum_{k=1}^n (\mathsf{m}_{ik}-\delta_{k,i}-\delta_{k,i+1})\varpi_k}\right)+\\ \sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\pm 1} b_i\cdots b_j v^{j-i+\sum_{i\leq a<b\leq j} (\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})+\sum_{k=1}^n\sum_{s=i}^j (k-n) \mathsf{m}_{sk}}\times\\ e^{-\alpha_i-\ldots-\alpha_j}T_{\sum_{k=1}^n(\sum_{s=i}^j \mathsf{m}_{sk}+\delta_{k,i}+\delta_{k,j+1})\varpi_k}+\\ \sum_{1\leq i<n-1}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-2,n}=\pm 1} b_i\cdots b_{n-2}b_n v^{n-i-1+\sum_{i\leq a<b\leq n}^{a,b\ne n-1} (\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})+\sum_{k=1}^n\sum_{i\leq s\leq n}^{s\ne n-1} (k-n) \mathsf{m}_{sk}}\times\\ e^{-\alpha_i-\ldots-\alpha_{n-2}-\alpha_n}T_{\sum_{k=1}^n(\sum_{i\leq s\leq n}^{s\ne n-1} \mathsf{m}_{sk}+\delta_{k,i}-\delta_{k,n})\varpi_k}+\\ \sum_{1\leq i<n-1}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-2,n-1}=\epsilon^{\pm}_{n-2,n}=\pm 1} b_i\cdots b_n v^{n-i+\sum_{i\leq a<b\leq n} (\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})+\sum_{k=1}^n \sum_{s=i}^n (k-n) \mathsf{m}_{sk}}\times \end{multline} (3.12) \begin{multline*} \\ e^{-\alpha_i-\ldots-\alpha_n}T_{\sum_{k=1}^n(\sum_{s=i}^{n} \mathsf{m}_{sk}+\delta_{k,i}-\delta_{k,n-1})\varpi_k}+\\ \sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\mp 1} b_i\cdots b_j v^{j-i+\sum_{i\leq a<b\leq j}(\mathsf{n}^+_{ba}-\mathsf{n}^-_{ba})+\sum_{k=1}^n \sum_{s=i}^j (k-n) \mathsf{m}_{sk}}\times\\ e^{-\alpha_i-\ldots-\alpha_j}T_{\sum_{k=1}^n(\sum_{s=i}^j \mathsf{m}_{sk}-\delta_{k,i}-\delta_{k,j+1})\varpi_k}+\\ \sum_{1\leq i<n-1}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-2,n}=\mp 1} b_i\cdots b_{n-2}b_n v^{n-i-1+\sum_{i\leq a<b\leq n}^{a,b\ne n-1} (\mathsf{n}^+_{ba}-\mathsf{n}^-_{ba})+\sum_{k=1}^n \sum_{i\leq s\leq n}^{s\ne n-1} (k-n) \mathsf{m}_{sk}}\times\\ e^{-\alpha_i-\ldots-\alpha_{n-2}-\alpha_n}T_{\sum_{k=1}^n(\sum_{i\leq s\leq n}^{s\ne n-1} \mathsf{m}_{sk}-\delta_{k,i}+\delta_{k,n})\varpi_k}+\\ \sum_{1\leq i<n-1}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-2,n-1}=\epsilon^{\pm}_{n-2,n}=\mp 1} b_i\cdots b_n v^{n-i+\sum_{i\leq a<b\leq n} (\mathsf{n}^+_{ba}-\mathsf{n}^-_{ba})+\sum_{k=1}^n \sum_{s=i}^n (k-n) \mathsf{m}_{sk}}\times\\ e^{-\alpha_i-\ldots-\alpha_n}T_{\sum_{k=1}^n(\sum_{s=i}^n \mathsf{m}_{sk}-\delta_{k,i}+\delta_{k,n-1})\varpi_k}. \end{multline*} Remark 3.16. If |$\epsilon ^+=\epsilon ^-$|⁠, then the last six sums are vacuous. If we further set |$\mathsf{n}^+=\mathsf{n}^-$| and |$c^{\pm }_i=\pm 1$| for all |$i$|⁠, then we obtain the formula for the 1st hamiltonian of the type |$D_n$| quantum difference Toda lattice as defined in [6] (we write down this formula as we could not find it in the literature, even though it can be derived completely analogously to [6, (5.7)], cf. [7, the end of Section 3]): \begin{multline} \textbf{D}_1= \sum_{i=1}^n (T_{2\varpi_i}+T_{-2\varpi_i})- (v-v^{-1})^2 \sum_{i=1}^{n-1} e^{-\alpha_i} \left(T_{\varpi_i+\varpi_{i+1}}+T_{-\varpi_i-\varpi_{i+1}}\right)-\\ (v-v^{-1})^2 e^{-\alpha_n}\left(T_{-\varpi_{n-1}+\varpi_n}+v^{-2} T_{\varpi_{n-1}-\varpi_n}\right)+ (v-v^{-1})^4 e^{-\alpha_{n-1}-\alpha_n}. \end{multline} (3.13) Recall the algebra |${\mathcal C}_n$| from Section 3.12. Consider the anti-isomorphism from |${\mathcal C}_n$| to the algebra |${\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{o}_{2n}})$| of Section 2.5, sending |${\mathsf{w}}_j\mapsto T_{-\varpi _j}, {\mathsf{D}}_i/{\mathsf{D}}_{i+1}\mapsto e^{-\alpha _i}, {\mathsf{D}}_n^2\mapsto e^{\alpha _{n-1}-\alpha _n}$|⁠. Let |$\mathsf{H}=\mathsf{H}(\epsilon ^{\pm }, \mathsf{n}^{\pm }, c^{\pm })$| be the element of |${\mathcal C}_n$| that corresponds to |$\textbf{D}_1$| under this anti-isomorphism (to save space, we omit the explicit long formula for |$\mathsf{H}$|⁠). The following is the key property of |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| in type |$D$|⁠. Proposition 3.17. |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| depends only on |$\vec{\epsilon }=(\epsilon _{n-1},\ldots ,\epsilon _1)\in \{-1,0,1\}^{n-1}$| with |$\epsilon _i:=\frac{\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}}{2}\ (1\leq i\leq n-2), \epsilon _{n-1}:=\frac{\epsilon ^+_{n-2,n}-\epsilon ^-_{n-2,n}}{2}$|⁠, up to algebra automorphisms of |${\mathcal C}_n$|⁠. The proof of this result is completely analogous to that of Proposition 3.11 given in Appendix A, see also Remark A.1; we leave the details to the interested reader. Proposition 3.17 implies that given two pairs of Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| and |$(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| with |$\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}= \tilde{\epsilon }^+_{i,i+1}-\tilde{\epsilon }^-_{i,i+1}\ (1\leq i\leq n-2)$| and |$\epsilon ^+_{n-2,n}-\epsilon ^-_{n-2,n}=\tilde{\epsilon }^+_{n-2,n}-\tilde{\epsilon }^-_{n-2,n}$|⁠, there exists an algebra automorphism of |${\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{o}_{2n}})$| that maps the 1st hamiltonian |$\textbf{D}_1(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |$\textbf{D}_1(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. As we will see in Appendix B, the same automorphism maps the modified quantum Toda system |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |${\mathcal{T}}(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. 3.18 Explicit formulas and classification in type |$B_n$| Recall explicit formulas for the action of |$U_v(\mathfrak{s}\mathfrak{o}_{2n+1})$| on its 1st fundamental representation |$V_1$|⁠. The space |$V_1$| has a basis |$\{w_0,\ldots ,w_{2n}\}$|⁠, in which the action is given via \begin{align*} & E_i(w_{2j-1})=\delta_{j,i+1}w_{2j-3},\ E_i(w_{2j})=\delta_{j,i}w_{2j+2},\ E_i(w_0)=0,\\ & F_i(w_{2j-1})=\delta_{j,i}w_{2j+1},\ F_i(w_{2j})=\delta_{j,i+1}w_{2j-2},\ F_i(w_0)=0,\\ & L_i(w_{2j-1})=v^{2\delta_{j\leq i}}w_{2j-1},\ L_i(w_{2j})=v^{-2\delta_{j\leq i}}w_{2j},\ L_i(w_0)=w_0,\\ & K_i(w_{2j-1})=v^{2\delta_{j,i}-2\delta_{j,i+1}}w_{2j-1},\ K_i(w_{2j})=v^{-2\delta_{j,i}+2\delta_{j,i+1}}w_{2j},\ K_i(w_0)=w_0,\\ & E_n(w_{2j-1})=0,\ E_n(w_{2j})=\delta_{j,n}w_{0},\ E_n(w_0)=w_{2n-1},\\ & F_n(w_{2j-1})=\delta_{j,n}w_{0},\ F_n(w_{2j})=0,\ F_n(w_0)=w_{2n},\\ & L_n(w_{2j-1})=vw_{2j-1},\ L_n(w_{2j})=v^{-1}w_{2j},\ L_n(w_0)=w_0,\\ & K_n(w_{2j-1})=v^{\delta_{j,n}}w_{2j-1},\ K_n(w_{2j})=v^{-\delta_{j,n}}w_{2j},\ K_n(w_0)=w_0 \end{align*} for any |$1\leq i<n, 1\leq j\leq n$|⁠. Let |$\varpi _j$| be the weight of |$w_{2j-1}\ (1\leq j\leq n)$|⁠, so that the weight of |$w_{2j}$| equals |$-\varpi _j$|⁠, while |$w_0$| has the zero weight. We note that now |$(\varpi _i,\varpi _j)=2\delta _{i,j}$|⁠. Recall that the simple roots are given by |$\alpha _i=\varpi _i-\varpi _{i+1}\ (1\leq i\leq n-1)$| and |$\alpha _n=\varpi _n$|⁠, while |$\rho =\sum _{i=1}^n (n+\frac{1}{2}-i)\varpi _i$| and |${\mathsf{d}}_1=\ldots ={\mathsf{d}}_{n-1}=2, {\mathsf{d}}_n=1$|⁠. To compute |$\textbf{D}_1$| explicitly, we use the same strategy as in type |$A$|⁠. In contrast to the types |$A,C,D$| treated above, |$E_n^2$| and |$F_n^2$| act nontrivially on |$V_1$|⁠, while |$\{E_i^r,F_i^r\}^{r>1}_{1\leq i<n}\cup \{E_n^s,F_n^s\}_{s>2}$| still act by zero on |$V_1$|⁠. Therefore, applying formula (3.1), we can replace |$R_{\alpha _i}$| by |$\bar{R}_{\alpha _i}:=1+(v^2-v^{-2})E_i\otimes F_i$| for |$1\leq i<n$| and |$R_{\alpha _n}$| by |$\bar{R}_{\alpha _n}:=1+(v-v^{-1})E_n\otimes F_n+cE_n^2\otimes F_n^2$| for |$c:=(1-v^{-1})(v-v^{-1})$|⁠. Let us now compute all the non-zero terms contributing to |$C^{\prime}_{V_1}$|⁠: Picking |$1$| out of each |$\bar{R}^{\textrm{op}}_i, \bar{R}_i$|⁠, we recover |$1+\sum _{i=1}^n \left (v^{4n+2-4i}\cdot K_{2\varpi _i}+v^{-4n-2+4i}\cdot\right. $||$ \left. K_{-2\varpi _i}\right )$|⁠. Picking nontrivial terms only at |$\bar{R}_{\alpha _j}^{\textrm{op}}, \bar{R}_{\alpha _i}$|⁠, the result does not depend on |$\textrm{Or}^{\pm }$| (hence, the orderings |$\prec _{\pm }$|⁠) and the total contribution of the nonzero terms equals \begin{align*} & \sum_{i=1}^{n-1} (v^2-v^{-2})^2\left(v^{4n+2-4i}F_iK_{\varpi_{i+1}}E_iK_{\varpi_i}+ v^{-4n+2+4i}F_iK_{-\varpi_i}E_iK_{-\varpi_{i+1}}\right)+\\ & (v-v^{-1})^2(F_nK_{-\varpi_n}E_n+v^2F_nE_nK_{\varpi_n})+ c^2v^2F_n^2K_{-\varpi_n}E_n^2K_{\varpi_n}. \end{align*} The contribution of the remaining terms to |$C^{\prime}_{V_1}$| depends on |$\textrm{Or}^{\pm }$|⁠. Tracing back explicit formulas for the action of |$U_v(\mathfrak{s}\mathfrak{o}_{2n+1})$| on |$V_1$|⁠, we see that the total sum of such terms equals \begin{align*} & \sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\pm 1} (v^2-v^{-2})^{2(j-i+1)}v^{4n+2-4i}\cdot F_i\cdots F_{j}K_{\varpi_{j+1}}E_{j}\cdots E_i K_{\varpi_i}+\\ & \sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\pm 1} (v^2-v^{-2})^{2(n-i)}(v-v^{-1})^2v^{4n+2-4i}\cdot F_i\cdots F_{n}E_{n}\cdots E_i K_{\varpi_i}+\\ & \sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\pm 1}\! c^2(v^2\!-\!v^{-2})^{2(n-i)}v^{4n+2-4i}\cdot F_i\cdots F_{n-1}F_n^2K_{-\varpi_n}E_{n}^2E_{n-1}\!\cdots E_i K_{\varpi_i}+\\ & \sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\mp 1} (v^2-v^{-2})^{2(j-i+1)}v^{-4n+2+4j}\cdot F_j\cdots F_{i}K_{-\varpi_{i}}E_{i}\cdots E_j K_{-\varpi_{j+1}}+\\ & \sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\mp 1} (v^2-v^{-2})^{2(n-i)}(v-v^{-1})^2\cdot F_n\cdots F_{i}K_{-\varpi_i}E_{i}\cdots E_n+\\ & \sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\mp 1} c^2(v^2-v^{-2})^{2(n-i)} v^2\cdot F_n^2F_{n-1}\cdots F_i K_{-\varpi_i}E_{i}\cdots E_{n-1}E_n^2 K_{\varpi_{n}}. \end{align*} We have listed all the nonzero terms contributing to |$C^{\prime}_{V_1}$|⁠. To obtain |$\tilde{\textbf{D}}_1=\bar{\textbf{D}}_{V_1}$|⁠, we should rewrite the above formulas via |$e_i,f_i$| and $$L_p=\begin{cases} K_{\varpi _1+\ldots +\varpi _p}, & \textrm{if}\ 1\leq p<n\\ K_{\frac{1}{2}(\varpi _1+\ldots +\varpi _n)}, & \textrm{if}\ p=n \end{cases},$$ moving all the Cartan terms to the middle, and then apply the characters |$\chi ^{\pm }$| with |$\chi ^+(e_i)=c^+_i, \chi ^-(f_i)=c^-_i$|⁠. Conjugating further by |$e^{\rho }$|⁠, we obtain the explicit formula for the 1st hamiltonian |$\textbf{D}_1$| of the type |$B_n$| modified quantum difference Toda system. To write it down, define constants |$b_i,\mathsf{m}_{ik}$| via |$\mathsf{m}_{ik}:=\sum _{p=k}^n(\mathsf{n}^-_{ip}-\mathsf{n}^+_{ip})(1-\frac{1}{2}\delta _{p,n})$| and |$b_i:=(v_i-v_i^{-1})^2v_i^{\mathsf{n}^+_{ii}-\mathsf{n}^-_{ii}}c^+_ic^-_i$|⁠. Then we have \begin{multline} \textbf{D}_1=1+ \sum_{i=1}^n (T_{2\varpi_i}+T_{-2\varpi_i})+\\[-4pt] b_n v^{\sum_{k=1}^n (2k-2n-1)\mathsf{m}_{nk}} \cdot e^{-\alpha_n} \left(vT_{\sum_{k=1}^n (\mathsf{m}_{nk}-\delta_{k,n})\varpi_k}+v^{-1}T_{\sum_{k=1}^n (\mathsf{m}_{nk}+\delta_{k,n})\varpi_k}\right)+\\ (1+v)^{-2}b_n^2 v^{-2+(\mathsf{n}_{nn}^+-\mathsf{n}_{nn}^-) + \sum_{k=1}^n (4k-4n-2)\mathsf{m}_{nk}} \cdot e^{-2\alpha_n}T_{\sum_{k=1}^n 2\mathsf{m}_{n,k}\varpi_k}+\\ \sum_{i=1}^{n-1} b_i v^{\sum_{k=1}^n (2k-2n-1)\mathsf{m}_{ik}}\cdot e^{-\alpha_i} \left(T_{\sum_{k=1}^n (\mathsf{m}_{ik}+\delta_{k,i}+\delta_{k,i+1})\varpi_k}+T_{\sum_{k=1}^n (\mathsf{m}_{ik}-\delta_{k,i}-\delta_{k,i+1})\varpi_k}\right)+\\[-4pt] \sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\pm 1} b_i\cdots b_j v^{2j-2i+2\sum_{i\leq a<b\leq j} (\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})+\sum_{k=1}^n \sum_{s=i}^j (2k-2n-1)\mathsf{m}_{sk}}\times\\ e^{-\alpha_i-\ldots-\alpha_j}T_{\sum_{k=1}^n(\sum_{s=i}^j \mathsf{m}_{sk}+\delta_{k,i}+\delta_{k,j+1})\varpi_k}+\\[-4pt] \sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\pm 1} b_i\cdots b_n v^{2n-2i-1+\sum_{i\leq a<b\leq n}(\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})(2-\delta_{b,n}) +\sum_{k=1}^n \sum_{s=i}^n (2k-2n-1)\mathsf{m}_{sk}}\times\\[-4pt]e^{-\alpha_i-\ldots-\alpha_n}T_{\sum_{k=1}^n(\sum_{s=i}^n \mathsf{m}_{sk}+\delta_{k,i})\varpi_k}+\\ \sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\pm 1} (1+v)^{-2}b_i\cdots b_{n-1}b_n^2 v^{2n-2i+(\mathsf{n}^+_{nn}-\mathsf{n}^-_{nn})+2\sum_{i\leq a<b\leq n} (\mathsf{n}^+_{ab}-\mathsf{n}^-_{ab})}\times\\ v^{\sum_{k=1}^n \sum_{s=i}^j (2k-2n-1)(1+\delta_{s,n})\mathsf{m}_{sk}}\cdot e^{-\alpha_i-\ldots-\alpha_{n-1}-2\alpha_n}T_{\sum_{k=1}^n(\sum_{s=i}^{n} \mathsf{m}_{sk}+\mathsf{m}_{nk}+\delta_{k,i}-\delta_{k,n})\varpi_k}+\\\sum_{1\leq i<j<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{j-1,j}=\mp 1} b_i\cdots b_j v^{2j-2i+2\sum_{i\leq a<b\leq j}(\mathsf{n}^+_{ba}-\mathsf{n}^-_{ba})+\sum_{k=1}^n \sum_{s=i}^j (2k-2n-1)\mathsf{m}_{sk}}\times\\ e^{-\alpha_i-\ldots-\alpha_j}T_{\sum_{k=1}^n(\sum_{s=i}^j \mathsf{m}_{sk}-\delta_{k,i}-\delta_{k,j+1})\varpi_k}+\\\sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\mp 1} b_i\cdots b_n v^{2n-2i+1+2\sum_{i\leq a<b\leq n} (\mathsf{n}^+_{ba}-\mathsf{n}^-_{ba})+\sum_{k=1}^n \sum_{s=i}^n (2k-2n-1)\mathsf{m}_{sk}}\times\\ e^{-\alpha_i-\ldots-\alpha_n}T_{\sum_{k=1}^n(\sum_{s=i}^n \mathsf{m}_{sk}-\delta_{k,i})\varpi_k}+\\ \sum_{1\leq i<n}^{\epsilon^{\pm}_{i,i+1}=\ldots=\epsilon^{\pm}_{n-1,n}=\mp 1} (1+v)^{-2}b_i\cdots b_{n-1}b_n^2 v^{2n-2i+(\mathsf{n}^+_{nn}-\mathsf{n}^-_{nn})+\sum_{i\leq a<b\leq n} (\mathsf{n}^+_{ba}-\mathsf{n}^-_{ba})(2+2\delta_{b,n})}\times\\ v^{\sum_{k=1}^n \sum_{s=i}^n (2k-2n-1)(1+\delta_{s,n})\mathsf{m}_{sk}}\cdot e^{-\alpha_i-\ldots-\alpha_{n-1}-2\alpha_n}T_{\sum_{k=1}^n(\sum_{s=i}^n \mathsf{m}_{sk}+\mathsf{m}_{nk}-\delta_{k,i}+\delta_{k,n})\varpi_k}. \end{multline} (3.14) Remark 3.19. If |$\epsilon ^+=\epsilon ^-$|⁠, then the last six sums are vacuous. If we further set |$\mathsf{n}^+=\mathsf{n}^-$| and |$c^{\pm }_i=\pm 1$| for all |$i$|⁠, then we obtain the formula for the 1st hamiltonian of the type |$B_n$| quantum difference Toda lattice as defined in [6] (we write down this formula as we could not find it in the literature, even though it can be derived completely analogously to [6, (5.7)], cf. [7, the end of Section 3]): \begin{multline} \textbf{D}_1= 1+\sum_{i=1}^n (T_{2\varpi_i}+T_{-2\varpi_i})- (v^2-v^{-2})^2 \sum_{i=1}^{n-1} e^{-\alpha_i} \left(T_{\varpi_i+\varpi_{i+1}}+T_{-\varpi_i-\varpi_{i+1}}\right)-\\ (v-v^{-1})^2 e^{-\alpha_n} \left(vT_{-\varpi_n}+v^{-1}T_{\varpi_n}\right)+ v^{-2}(1-v^{-1})^2(v-v^{-1})^2 e^{-2\alpha_n}. \end{multline} (3.15) Recall the algebra |${\mathcal{A}}_n$| from Section 3.9. Consider the anti-isomorphism from |${\mathcal{A}}_n$| to the algebra |${\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{o}_{2n+1}})$| of Section 2.5, sending |${\mathsf{w}}_j\mapsto T_{-\varpi _j}, {\mathsf{D}}_j\mapsto e^{-\sum _{k=j}^n \alpha _k}$|⁠. Let |$\mathsf{H}=\mathsf{H}(\epsilon ^{\pm }, \mathsf{n}^{\pm }, c^{\pm })$| be the element of |${\mathcal{A}}_n$| that corresponds to |$\textbf{D}_1$| under this anti-isomorphism (to save space, we omit the explicit long formula for |$\mathsf{H}$|⁠). The following is the key property of |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| in type |$B$|⁠. Proposition 3.20. |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| depends only on |$\vec{\epsilon }=(\epsilon _{n-1},\ldots ,\epsilon _1)\in \{-1,0,1\}^{n-1}$| with |$\epsilon _i:=\frac{\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}}{2}$|⁠, up to algebra automorphisms of |${\mathcal{A}}_n$|⁠. The proof of this result is analogous to that of Proposition 3.11 given in Appendix A, see also Remark A.1; we leave the details to the interested reader. Proposition 3.20 implies that given two pairs of Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| and |$(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| with |$\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}=\tilde{\epsilon }^+_{i,i+1}-\tilde{\epsilon }^-_{i,i+1}$|⁠, there exists an algebra automorphism of |${\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{o}_{2n+1}})$| that maps the 1st hamiltonian |$\textbf{D}_1(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |$\textbf{D}_1(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. As we will see in Appendix B, the same automorphism maps the modified quantum Toda system |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |${\mathcal{T}}(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. 3.21 Lax matrix realization in type |$A$| In this section, we identify the type |$A_{n-1}$| modified quantum difference Toda systems with those discovered in [10, 11(ii, iii)] via the Lax matrix formalism. Recall the algebra |${\mathcal{A}}_n$| of Section 3.9. Consider the following three (local) Lax matrices: \begin{equation} L^{v,-1}_i(z):=\left( \begin{array}{cc} {\mathsf{w}}_i^{-1}-{\mathsf{w}}_iz^{-1} & {\mathsf{w}}_i {\mathsf{D}}_i^{-1} \\ -{\mathsf{w}}_i{\mathsf{D}}_iz^{-1} & {\mathsf{w}}_i \\ \end{array} \right)\in \textrm{Mat}(2, z^{-1}{\mathcal{A}}_n[z]), \end{equation} (3.16) \begin{equation} L^{v,0}_i(z):= \left( \begin{array}{cc} {\mathsf{w}}_i^{-1}z^{1/2}-{\mathsf{w}}_i z^{-1/2} & {\mathsf{D}}^{-1}_iz^{1/2} \\ -{\mathsf{D}}_iz^{-1/2} & 0 \\ \end{array} \right)\in{\operatorname{Mat}}(2,z^{-1/2}{\mathcal{A}}_n[z]), \end{equation} (3.17) \begin{equation} L^{v,1}_i(z):=\left( \begin{array}{cc} {\mathsf{w}}_i^{-1}z-{\mathsf{w}}_i & {\mathsf{w}}_i^{-1}{\mathsf{D}}_i^{-1}z \\ -{\mathsf{w}}_i^{-1}{\mathsf{D}}_i & -{\mathsf{w}}_i^{-1} \\ \end{array} \right)\in \textrm{Mat}(2, {\mathcal{A}}_n[z]). \end{equation} (3.18) For any |$\vec{k}=(k_n,\ldots ,k_1)\in \{-1,0,1\}^n$|⁠, define the mixed complete monodromy matrix \begin{equation} {\mathsf{T}}^v_{\vec{k}}(z):=L^{v,k_n}_n(z)\cdots L^{v,k_1}_{1}(z). \end{equation} (3.19) We also recall the standard trigonometric |$R$|-matrix \begin{equation*} R_{\textrm{trig}}(z):=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \frac{z-1}{vz-v^{-1}} & \frac{z(v-v^{-1})}{vz-v^{-1}} & 0 \\ 0 & \frac{v-v^{-1}}{vz-v^{-1}} & \frac{z-1}{vz-v^{-1}} & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right). \end{equation*} The following key property of the complete monodromy matrices is established in [10, 11(ii)]. Proposition 3.22. For any |$\vec{k}\!\in\! \{-1,0,1\}^n$|⁠, |${\mathsf{T}}^v_{\vec{k}}(z)$| satisfies the trigonometric RTT relation: \begin{equation*} R_{\textrm{trig}}(z/w)\left({\mathsf{T}}^v_{\vec{k}}(z)\otimes 1\right)\left(1\otimes{\mathsf{T}}^v_{\vec{k}}(w)\right)= \left(1\otimes{\mathsf{T}}^v_{\vec{k}}(w)\right)\left({\mathsf{T}}^v_{\vec{k}}(z)\otimes 1\right)R_{\textrm{trig}}(z/w). \end{equation*} As an immediate corollary of this result, we obtained (see [10, 11(iii)]) the following. Proposition 3.23. ([10]). Fix |$\vec{k}=(k_n,\ldots ,k_1)\in \{-1,0,1\}^n$|⁠. (a) The coefficients in powers of |$z$| of the matrix element |${\mathsf{T}}^v_{\vec{k}}(z)_{11}$| generate a commutative subalgebra of |${\mathcal{A}}_n$|⁠. Moreover, they lie in the subalgebra of |${\mathcal{A}}_n$| generated by |$\{{\mathsf{w}}_j^{\pm 1}, ({\mathsf{D}}_i/{\mathsf{D}}_{i+1})^{\pm 1}\}_{1\leq i<n}^{1\leq j\leq n}$|⁠. (b) |${\mathsf{T}}^v_{\vec{k}}(z)_{11}= (-1)^n{\mathsf{w}}_1\cdots{\mathsf{w}}_n\left (z^s-{\mathsf H}^{\vec{k}}_2z^{s+1}+\textrm{higher powers of}\ z\right ),$| where |$s=\sum _{j=1}^n \frac{k_j-1}{2}$|⁠. The hamiltonian |${\mathsf H}^{\vec{k}}_2$| equals \begin{equation} {\mathsf H}^{\vec{k}}_2= \sum_{j=1}^n {\mathsf{w}}^{-2}_j+ \sum_{i=1}^{n-1} {\mathsf{w}}_i^{-k_i-1}{\mathsf{w}}_{i+1}^{-k_{i+1}-1}\cdot \frac{{\mathsf{D}}_i}{{\mathsf{D}}_{i+1}}+ \sum_{1\leq i<j-1\leq n-1}^{k_{i+1}=\ldots=k_{j-1}=1} {\mathsf{w}}_i^{-k_i-1}\cdots{\mathsf{w}}_j^{-k_j-1}\cdot \frac{{\mathsf{D}}_i}{{\mathsf{D}}_j}. \end{equation} (3.20) (c) Set |$\vec{k}^{\prime}=(0,k_{n-1},\ldots ,k_2,0)$|⁠. Then |${\mathsf H}^{\vec{k}}_2$| is conjugate to |${\mathsf H}^{\vec{k}^{\prime}}_2$|⁠. Let |${\mathcal{T}}^{\vec{k}}$| denote the commutative subalgebra of |$\bar{{\mathcal{A}}}_n$| generated by the images of the coefficients in powers of |$z$| of the matrix element |$ {\mathsf{T}}^v_{\vec{k}}(z)_{11}$|⁠, while |$\bar{\mathsf{H}}^{\vec{k}}_2\in{\mathcal{T}}^{\vec{k}}$| denote the image of |$\mathsf{H}^{\vec{k}}_2$|⁠. The main result of this section identifies |${\mathcal{T}}^{\vec{k}}$| with the pre-images |$\widetilde{{\mathcal{T}}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| of |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| in type |$A_{n-1}$| under the anti-isomorphism |$\bar{{\mathcal{A}}}_n\to{\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{l}_n})$| of Section 3.9. This provides a Lax matrix realization of the type |$A$| modified quantum difference Toda systems. Theorem 3.24. Given a pair of type |$A_{n-1}$| Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| and |$\vec{k}\in \{-1,0,1\}^n$| satisfying |$k_{i+1}=\frac{\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}}{2}$| for any |$1\leq i\leq n-2$|⁠, the following holds: (a) There is an algebra automorphism of |$\bar{{\mathcal{A}}}_n$| that maps |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |$\bar{\mathsf{H}}^{\vec{k}}_2$|⁠. (b) The automorphism of part (a) maps |$\widetilde{{\mathcal{T}}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |${\mathcal{T}}^{\vec{k}}$|⁠. The proof of Theorem 3.24 is presented in Appendix E and closely follows our proofs of Proposition 3.11 (see Appendix A) and Theorem 3.2 (see Appendix B). Remark 3.25. For |$\vec{k}=\vec{0}$|⁠, we recover the Lax matrix realization of the type |$A$| quantum difference Toda lattice, due to [15]. Actually, the above construction admits a standard one-parameter deformation of commutative subalgebras of |${\mathcal{A}}_n$| as provided by the following result. Proposition 3.26. (a) For any |$\varepsilon \in{\mathbb{C}}(v^{1/\mathsf{N}})$|⁠, the coefficients in powers of |$z$| of the linear combination |${\mathsf{T}}^v_{\vec{k}}(z)_{11}+\varepsilon{\mathsf{T}}^v_{\vec{k}}(z)_{22}$| generate a commutative subalgebra of |${\mathcal{A}}_n$|⁠. Moreover, they lie in the subalgebra of |${\mathcal{A}}_n$| generated by |$\{{\mathsf{w}}_j^{\pm 1}, ({\mathsf{D}}_i/{\mathsf{D}}_{i+1})^{\pm 1}\}_{1\leq i<n}^{1\leq j\leq n}$|⁠. (b) We have |${\mathsf{T}}^v_{\vec{k}}(z)_{11}+\varepsilon{\mathsf{T}}^v_{\vec{k}}(z)_{22}= (-1)^n{\mathsf{w}}_1\cdots{\mathsf{w}}_n\left (\hat{\mathsf{H}}^{\vec{k}}_1 z^s-\hat{\mathsf{H}}^{\vec{k}}_2 z^{s+1}+\textrm{higher powers of}\ z\right )$|⁠, where |$s=\sum _{j=1}^n \frac{k_j-1}{2}$|⁠. Here |$\hat{\mathsf{H}}^{\vec{k}}_1=1+\varepsilon \prod _{j=1}^n \delta _{k_j,1}\prod _{j=1}^n {\mathsf{w}}_j^{-2}$|⁠, while |$\hat{\mathsf{H}}^{\vec{k}}_2$| is given by \begin{multline} \hat{\mathsf{H}}^{\vec{k}}_2= \sum_{j=1}^n {\mathsf{w}}^{-2}_j+ \sum_{i=1}^{n-1} {\mathsf{w}}_i^{-k_i-1}{\mathsf{w}}_{i+1}^{-k_{i+1}-1}\cdot \frac{{\mathsf{D}}_i}{{\mathsf{D}}_{i+1}}+ \sum_{1\leq i<j-1\leq n-1}^{k_{i+1}=\ldots=k_{j-1}=1} {\mathsf{w}}_i^{-k_i-1}\cdots{\mathsf{w}}_j^{-k_j-1}\cdot \frac{{\mathsf{D}}_i}{{\mathsf{D}}_j}+\\ \varepsilon \left(\sum_{1\leq i<j\leq n}^{\substack{k_1=\ldots=k_{i-1}=1\\k_{j+1}=\ldots=k_n=1}} {\mathsf{w}}_1^{-k_1-1}\cdots{\mathsf{w}}_i^{-k_i-1}{\mathsf{w}}_j^{-k_j-1}\cdots{\mathsf{w}}_n^{-k_n-1}\cdot \frac{{\mathsf{D}}_j}{{\mathsf{D}}_i}+ \sum_{1\leq j\leq n}^{\substack{k_j=-1\\k_{i}=1 (i\ne j)}} \prod_{k\ne j} {\mathsf{w}}_k^{-2}\right). \end{multline} (3.21) Proof. (a) Follows from the equality |$\left [{\mathsf{T}}^v_{\vec{k}}(z)_{11}+\varepsilon{\mathsf{T}}^v_{\vec{k}}(z)_{22}, {\mathsf{T}}^v_{\vec{k}}(w)_{11}+\varepsilon{\mathsf{T}}^v_{\vec{k}}(w)_{22}\right ]=0$|⁠, which is implied by the RTT relation of Proposition 3.22. (b) Straightforward computation. Definition 3.27. A type |$A_{n-1}$| periodic modified quantum difference Toda system is the commutative subalgebra |$\hat{{\mathcal{T}}}^{\vec{k},\varepsilon }$| of |$\bar{{\mathcal{A}}}_n$|⁠. We note that |$\hat{{\mathcal{T}}}^{\vec{k},0}$| coincides with |${\mathcal{T}}^{\vec{k}}$|⁠. Remark 3.28. (a) In particular, |$\hat{\mathsf{H}}^{\vec{0}}_2$| is conjugate (in the sense of (A1, A2)) to the element of |$\bar{{\mathcal{A}}}_n$| that corresponds under the anti-isomorphism |$\bar{{\mathcal{A}}}_n\to{\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{l}_n})$| of Section 3.9 to the 1st hamiltonian of the type |$A^{(1)}_{n-1}$| quantum difference affine Toda system of [6, (5.9)]: \begin{equation} \hat{\textbf{D}}_1=\sum_{j=1}^n T_{2\varpi_j}- (v-v^{-1})^2\sum_{i=1}^{n-1} e^{-\varpi_i+\varpi_{i+1}}T_{\varpi_i+\varpi_{i+1}}- \kappa(v-v^{-1})^2e^{-\varpi_n+\varpi_1}T_{\varpi_n+\varpi_1} \end{equation} (3.22) with |$\kappa =(-1)^n(v-v^{-1})^{-2n}\varepsilon $|⁠. The quantum difference affine Toda systems are defined similarly to the (finite type) |$q$|-Toda systems of loc.cit., but starting from a quantum affine algebra and its center at the critical level. The parameter |$\kappa \in{\mathbb{C}}(v^{1/\mathsf{N}})$| is essential, that is, it cannot be removed, cf. [6, Remark 1]. (b) We expect that most of our results from this paper can be generalized to an affine setting. In particular, the type |$A_{n-1}$| periodic modified quantum difference Toda systems introduced above should be conjugate to the type |$A_{n-1}^{(1)}$| modified quantum difference affine Toda systems, thus generalizing part (a) of the current Remark. To state this more precisely, let us first specify what we mean by a Sevostyanov triple |$(\hat{\epsilon },\hat{\mathsf{n}},\hat{c})$| for |$\hat{\mathfrak{g}}$| of an affine type, except |$A^{(1)}_1$|⁠. Let |$\alpha _0,\ldots ,\alpha _n$| be simple positive roots of |$\hat{\mathfrak{g}}$| (with |$\alpha _0$| the distinguished one) and |$\{a_i\}_{i=0}^n$| be the labels on the Dynkin diagram of |$\hat{\mathfrak{g}}$| as in [12, Chapter IV, Table Aff]. Then a Sevostyanov triple |$(\hat{\epsilon },\hat{\mathsf{n}},\hat{c})$| is a collection of the following data: (1) |$\hat{\epsilon }=(\epsilon _{ij})_{i,j=0}^n$| is the associated matrix of an orientation of |$\textrm{Dyn}(\hat{\mathfrak{g}})$| as before, (2) |$\hat{c}=(c_i)_{i=0}^n\in ({\mathbb{C}}(v^{1/\mathsf{N}})^{\times })^{n+1}$| is a collection of nonzero constants, and (3) |$\hat{\mathsf{n}}=(\mathsf{n}_{ij})_{0\leq i\leq n}^{1\leq j\leq n}$| is an integer matrix satisfying |${\mathsf{d}}_j\mathsf{n}_{ij}-{\mathsf{d}}_i\mathsf{n}_{ji}=\epsilon _{ij}b_{ij}$| for |$1\leq i,j\leq n$| and |${\mathsf{d}}_j\mathsf{n}_{0j}+\sum _{p=1}^n{\mathsf{d}}_p\mathsf{n}_{jp}\frac{a_p}{a_0}=\epsilon _{0j}b_{0j}$| for |$1\leq j\leq n$|⁠. In particular, given a pair of type |$A^{(1)}_{n-1}\ (n>2)$| Sevostyanov triples |$(\hat{\epsilon }^{\pm },\hat{\mathsf{n}}^{\pm },\hat{c}^{\pm })$|⁠, the corresponding difference operator |$\mathsf{H}(\hat{\epsilon }^{\pm },\hat{\mathsf{n}}^{\pm },\hat{c}^{\pm })\in \bar{{\mathcal{A}}}_n$| should depend (up to algebra automorphisms of |$\bar{{\mathcal{A}}}_n$|⁠) only on |$\vec{\epsilon }=(\epsilon _{n-1},\ldots ,\epsilon _0)\in \{-1,0,1\}^n$|⁠, where |$\epsilon _i:=\frac{\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}}{2}$| with indices considered modulo |$n$|⁠. The resulting |$3^n$| difference operators |$\mathsf{H}(\hat{\epsilon }^{\pm },\hat{\mathsf{n}}^{\pm },\hat{c}^{\pm })$| should be conjugate to the images of |$3^n$| hamiltonians |$\hat{\mathsf{H}}^{\vec{k}}_2$| in |$\bar{{\mathcal{A}}}_n$|⁠. We have verified this result for |$\epsilon ^{\pm }_{0,1}=\epsilon ^{\pm }_{0,n-1}=1$|⁠. (c) The type |$A^{(1)}_1$| deserves a special treatment, since it is the only affine type for which the analogue of Lemma 2.4 fails to hold (as |$a_{01}=a_{10}=-2$|⁠). Instead, such characters |$\chi ^{\pm }$| exist if and only if |$\mathsf{n}^{\pm }_{01}+\mathsf{n}^{\pm }_{11}\in \{-2,0,2\}$|⁠. Let |$\mathsf{H}(\hat{\mathsf{n}}^{\pm },\hat{c}^{\pm })$| be the element of |$\bar{{\mathcal{A}}}_2$| that corresponds to the 1st hamiltonian of the type |$A^{(1)}_1$| modified quantum difference affine Toda system associated with the pair |$(\hat{\mathsf{n}}^{\pm },\hat{c}^{\pm })$|⁠. We expect that |$\mathsf{H}(\hat{\mathsf{n}}^{\pm },\hat{c}^{\pm })$| depends (up to algebra automorphisms of |$\bar{{\mathcal{A}}}_2$|⁠) only on |$\frac{\mathsf{n}^+_{01}+\mathsf{n}^+_{11}-\mathsf{n}^-_{01}-\mathsf{n}^-_{11}}{2}\in \{\pm 2,\pm 1,0\}$|⁠. On the other hand, it is easy to see that the equivalence class of the difference operator |$\hat{\mathsf{H}}^{(k_2,k_1)}_2$| depends only on |$k_2-k_1\in \{\pm 2,\pm 1,0\}$|⁠. We expect that the resulting five difference operators in |$\bar{{\mathcal{A}}}_2$| are conjugate to the aforementioned five difference operators |$\mathsf{H}(\hat{\mathsf{n}}^{\pm },\hat{c}^{\pm })$|⁠. 3.29 Lax matrix realization in type |$C$| Motivated by the construction of the previous section, we provide a Lax matrix realization of the type |$C$| modified quantum difference Toda systems. In addition to |$L^{v,k}_i(z)\ (k=\pm 1,0)$| of (3.16–3.18), consider three more (local) Lax matrices: \begin{equation} \bar{L}^{v,-1}_i(z):=\left( \begin{array}{cc} {\mathsf{w}}_i-{\mathsf{w}}_i^{-1}z^{-1} & {\mathsf{w}}_i^{-1}{\mathsf{D}}_i \\ -{\mathsf{w}}_i^{-1}{\mathsf{D}}_i^{-1}z^{-1} & {\mathsf{w}}_i^{-1} \\ \end{array} \right)\in{\operatorname{Mat}}(2,z^{-1}{\mathcal{A}}_n[z]), \end{equation} (3.23) \begin{equation} \bar{L}^{v,0}_i(z):= \left( \begin{array}{cc} {\mathsf{w}}_i z^{1/2}-{\mathsf{w}}_i^{-1} z^{-1/2} & {\mathsf{D}}_iz^{1/2} \\ -{\mathsf{D}}_i^{-1}z^{-1/2} & 0 \\ \end{array} \right)\in{\operatorname{Mat}}(2,z^{-1/2}{\mathcal{A}}_n[z]), \end{equation} (3.24) \begin{equation} \bar{L}^{v,1}_i(z):=\left( \begin{array}{cc} {\mathsf{w}}_iz-{\mathsf{w}}_i^{-1} & {\mathsf{w}}_i{\mathsf{D}}_iz \\ -{\mathsf{w}}_i{\mathsf{D}}_i^{-1} & -{\mathsf{w}}_i \\ \end{array} \right)\in{\operatorname{Mat}}(2,{\mathcal{A}}_n[z]). \end{equation} (3.25) For any |$\vec{k}=(k_n,\ldots ,k_1)\in \{-1,0,1\}^n$|⁠, define the double mixed complete monodromy matrix \begin{equation} {\mathbb{T}}^v_{\vec{k}}(z):= \bar{L}^{v,-k_1}_1(z)\cdots \bar{L}^{v,-k_n}_{n}(z)L^{v,k_n}_n(z)\cdots L^{v,k_1}_{1}(z). \end{equation} (3.26) Let us summarize the key properties of the double mixed complete monodromy matrices. Proposition 3.30. Fix |$\vec{k}=(k_n,\ldots ,k_1)\in \{-1,0,1\}^n$|⁠. (a) |${\mathbb{T}}^v_{\vec{k}}(z)$| satisfies the trigonometric RTT relation \begin{equation*} R_{\textrm{trig}}(z/w)\left({\mathbb{T}}^v_{\vec{k}}(z)\otimes 1\right)\left(1\otimes{\mathbb{T}}^v_{\vec{k}}(w)\right)= \left(1\otimes{\mathbb{T}}^v_{\vec{k}}(w)\right)\left({\mathbb{T}}^v_{\vec{k}}(z)\otimes 1\right)R_{\textrm{trig}}(z/w). \end{equation*} (b) The coefficients in powers of |$z$| of the matrix element |${\mathbb{T}}^v_{\vec{k}}(z)_{11}$| generate a commutative subalgebra of |${\mathcal{A}}_n$|⁠. Moreover, they belong to the subalgebra |${\mathcal C}_n$| of |${\mathcal{A}}_n$|⁠, introduced in Section 3.12. (c) We have |${\mathbb{T}}^v_{\vec{k}}(z)_{11}=z^{-n}-{\mathbb{H}}^{\vec{k}}_2z^{-n+1}+\textrm{higher powers of}\ z$|⁠. The hamiltonian |${\mathbb{H}}^{\vec{k}}_2$| equals \begin{multline} {\mathbb{H}}^{\vec{k}}_2= \sum_{i=1}^n ({\mathsf{w}}_i^2+{\mathsf{w}}_i^{-2})+ \sum_{i=1}^{n-1} \left({\mathsf{w}}_i^{-k_i-1}{\mathsf{w}}_{i+1}^{-k_{i+1}-1}+{\mathsf{w}}_i^{-k_i+1}{\mathsf{w}}_{i+1}^{-k_{i+1}+1}\right) \cdot \frac{{\mathsf{D}}_i}{{\mathsf{D}}_{i+1}}+v^{-k_n}{\mathsf{w}}_n^{-2k_n}\cdot{\mathsf{D}}_n^2+\\ \sum_{1\leq i<j< n}^{k_{i+1}=\ldots=k_j=1} {\mathsf{w}}_i^{-k_i-1}\cdots{\mathsf{w}}_{j+1}^{-k_{j+1}-1}\cdot \frac{{\mathsf{D}}_i}{{\mathsf{D}}_{j+1}}+ \sum_{1\leq i<n}^{k_{i+1}=\ldots=k_n=1} v^{-1}{\mathsf{w}}_i^{-k_i-1}\cdots{\mathsf{w}}_n^{-k_n-1}\cdot{\mathsf{D}}_i{\mathsf{D}}_n+\\ \sum_{1\leq i<j< n}^{k_{i+1}=\ldots=k_j=-1} {\mathsf{w}}_i^{-k_i+1}\cdots{\mathsf{w}}_{j+1}^{-k_{j+1}+1}\cdot \frac{{\mathsf{D}}_i}{{\mathsf{D}}_{j+1}}+ \sum_{1\leq i<n}^{k_{i+1}=\ldots=k_n=-1} v {\mathsf{w}}_i^{-k_i+1}\cdots{\mathsf{w}}_n^{-k_n+1}\cdot{\mathsf{D}}_i{\mathsf{D}}_n. \end{multline} (3.27) Proof. (a) Note that |$\bar{L}^{v,k}_i(z)$| is obtained from |$L^{v,k}_i(z)$| by applying the automorphism of |${\mathcal{A}}_n$| that maps |${\mathsf{w}}_i^{\pm 1}\mapsto{\mathsf{w}}_i^{\mp 1}, {\mathsf{D}}_i^{\pm 1}\mapsto{\mathsf{D}}_i^{\mp 1}$|⁠. Hence, each of them satisfies the trigonometric RTT relation. Thus, an arbitrary product of |$L^{v,k}_i(z)$| and |$\bar{L}^{v,k}_i(z)$| also satisfies the trigonometric RTT relation. (b) This is an immediate corollary of (a). (c) Straightforward computation. Let |${\mathcal{T}}^{\vec{k}}$| denote the commutative subalgebra of |${\mathcal C}_n$| generated by the coefficients in powers of |$z$| of the matrix element |${\mathbb{T}}^v_{\vec{k}}(z)_{11}$|⁠. The main result of this section identifies |${\mathcal{T}}^{\vec{k}}$| with the pre-images |$\widetilde{{\mathcal{T}}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| of |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| in type |$C_n$| under the anti-isomorphism |${\mathcal C}_n\to{\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{p}_{2n}})$| of Section 3.12. This provides a Lax matrix realization of the type |$C$| modified quantum difference Toda systems. Theorem 3.31. Given a pair of type |$C_n$| Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| and |$\vec{k}\in \{-1,0,1\}^n$| satisfying |$k_{i+1}=\frac{\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}}{2}$| for any |$1\leq i\leq n-1$|⁠, the following holds: (a) There is an algebra automorphism of |${\mathcal C}_n$| which maps |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |${\mathbb{H}}^{\vec{k}}_2$|⁠. (b) The automorphism of part (a) maps |$\widetilde{{\mathcal{T}}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |${\mathcal{T}}^{\vec{k}}$|⁠. The proof of Theorem 3.31 is completely analogous to that of Theorem 3.24 given in Appendix E, see also Remark A.1; we leave the details to the interested reader. Remark 3.32. Recall that any nonsimply laced simple Lie algebra |$\mathfrak{g}^{\prime}$| admits a folding realization|$\mathfrak{g}^{\prime}\simeq \mathfrak{g}^{\sigma }$|⁠, where |$\mathfrak{g}$| is a simply laced Lie algebra endowed with an outer automorphism |$\sigma $| of a finite order (arising as an automorphism of the corresponding Dynkin diagram). This observation allows to relate the classical Toda system of |$\mathfrak{g}^{\prime}$| to the Toda system of |$\mathfrak{g}$|⁠; see [17]. The above construction of the modified quantum difference Toda systems in types |$C_n$| and |$A_{2n-1}$| via Lax matrices exhibits the former as a folding of the latter, once we require that the orientations |$\epsilon ^{\pm }$| of |$\textrm{Dyn}(\mathfrak{s}\mathfrak{l}_{2n})=A_{2n-1}$| satisfy |$\epsilon ^{\pm }_{i,i+1}=-\epsilon ^{\pm }_{2n-i-1,2n-i}$| for all |$i$|⁠. However, such a naive approach fails to work for the pair |$(B_n,D_{n+1})$|⁠. It would be interesting to understand the explicit relation. Let us warn the interested reader that the folding for quantum groups is more elaborate than in the classical set-up; see [1]. Analogously to the type |$A$| case, the above construction admits a standard one-parameter deformation of commutative subalgebras of |${\mathcal C}_n$| as provided by the following result. Proposition 3.33. (a) For any |$\varepsilon \in{\mathbb{C}}(v^{1/\mathsf{N}})$|⁠, the coefficients in powers of |$z$| of the linear combination |${\mathbb{T}}^v_{\vec{k}}(z)_{11}+\varepsilon{\mathbb{T}}^v_{\vec{k}}(z)_{22}$| generate a commutative subalgebra of |${\mathcal{A}}_n$|⁠. Moreover, they belong to the subalgebra |${\mathcal C}_n$| of |${\mathcal{A}}_n$|⁠. (b) We have |${\mathbb{T}}^v_{\vec{k}}(z)_{11}+\varepsilon{\mathbb{T}}^v_{\vec{k}}(z)_{22}= z^{-n}-\hat{{\mathbb{H}}}^{\vec{k}}_2 z^{-n+1}+\textrm{higher powers of}\ z$|⁠. The hamiltonian |$\hat{{\mathbb{H}}}^{\vec{k}}_2$| is given by the following formula: \begin{multline*} \hat{{\mathbb{H}}}^{\vec{k}}_2= \sum_{i=1}^n ({\mathsf{w}}_i^2+{\mathsf{w}}_i^{-2})+ \sum_{i=1}^{n-1} \left({\mathsf{w}}_i^{-k_i-1}{\mathsf{w}}_{i+1}^{-k_{i+1}-1}+{\mathsf{w}}_i^{-k_i+1}{\mathsf{w}}_{i+1}^{-k_{i+1}+1}\right) \cdot \frac{{\mathsf{D}}_i}{{\mathsf{D}}_{i+1}}+v^{-k_n}{\mathsf{w}}_n^{-2k_n}\cdot{\mathsf{D}}_n^2+\\ \sum_{1\leq i<j< n}^{k_{i+1}=\ldots=k_j=1} {\mathsf{w}}_i^{-k_i-1}\cdots{\mathsf{w}}_{j+1}^{-k_{j+1}-1}\cdot \frac{{\mathsf{D}}_i}{{\mathsf{D}}_{j+1}}+ \sum_{1\leq i<n}^{k_{i+1}=\ldots=k_n=1} v^{-1}{\mathsf{w}}_i^{-k_i-1}\cdots{\mathsf{w}}_n^{-k_n-1}\cdot{\mathsf{D}}_i{\mathsf{D}}_n+\\ \end{multline*} \begin{multline} \sum_{1\leq i<j< n}^{k_{i+1}=\ldots=k_j=-1} {\mathsf{w}}_i^{-k_i+1}\cdots{\mathsf{w}}_{j+1}^{-k_{j+1}+1}\cdot \frac{{\mathsf{D}}_i}{{\mathsf{D}}_{j+1}}+ \sum_{1\leq i<n}^{k_{i+1}=\ldots=k_n=-1} v {\mathsf{w}}_i^{-k_i+1}\cdots{\mathsf{w}}_n^{-k_n+1}\cdot{\mathsf{D}}_i{\mathsf{D}}_n+\\ \varepsilon\left(\delta_{n,1}(1-\delta_{k_1,0}){\mathsf{w}}_1^{-2k_1}+v^{k_1}{\mathsf{w}}_1^{-2k_1}\cdot{\mathsf{D}}_1^{-2}+ \prod_{i=1}^n \delta_{k_i,1}\cdot \prod_{i=1}^n {\mathsf{w}}_i^{-2}\cdot \frac{{\mathsf{D}}_n}{{\mathsf{D}}_1}+ \prod_{i=1}^n \delta_{k_i,-1}\cdot \prod_{i=1}^n {\mathsf{w}}_i^{2}\cdot \frac{{\mathsf{D}}_n}{{\mathsf{D}}_1}+\right.\\ \left.\sum_{1\leq i<n}^{k_1=\ldots=k_i=1} v {\mathsf{w}}_1^{-k_1-1}\cdots{\mathsf{w}}_{i+1}^{-k_{i+1}-1}\cdot \frac{1}{{\mathsf{D}}_1{\mathsf{D}}_{i+1}}+ \sum_{1\leq i<n}^{k_1=\ldots=k_i=-1} v^{-1} {\mathsf{w}}_1^{-k_1+1}\cdots{\mathsf{w}}_{i+1}^{-k_{i+1}+1}\cdot \frac{1}{{\mathsf{D}}_1{\mathsf{D}}_{i+1}}\right). \end{multline} (3.28) Definition 3.34. A type |$C_n$| periodic modified quantum difference Toda system is the commutative subalgebra |$\hat{{\mathcal{T}}}^{\vec{k},\varepsilon }$| of |${\mathcal C}_n$|⁠. We note that |$\hat{{\mathcal{T}}}^{\vec{k},0}$| coincides with |${\mathcal{T}}^{\vec{k}}$|⁠. Remark 3.35. (a) In particular, |$\hat{{\mathbb{H}}}^{\vec{0}}_2$| is conjugate to the element of |${\mathcal C}_n$| that corresponds under the anti-isomorphism |${\mathcal C}_n\to{\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{p}_{2n}})$| of Section 3.12 to the 1st hamiltonian of the type |$C^{(1)}_{n}$| quantum difference affine Toda system of [6] (cf. Remark 3.28(a)): \begin{multline} \hat{\textbf{D}}_1=\sum_{i=1}^n (T_{2\varpi_i}+T_{-2\varpi_i})- (v-v^{-1})^2\sum_{i=1}^{n-1} e^{-\varpi_i+\varpi_{i+1}}\left(T_{\varpi_i+\varpi_{i+1}}+T_{-\varpi_i-\varpi_{i+1}}\right)-\\ (v^2-v^{-2})^2 e^{-2\varpi_n}-\kappa v^{-2n-2}(v^2-v^{-2})^2e^{2\varpi_1} \end{multline} (3.29) with |$\kappa =v^{2n+2}(v-v^{-1})^{-4(n-1)}(v^2-v^{-2})^{-4}\varepsilon $|⁠. Here |$\kappa \in{\mathbb{C}}(v^{1/\mathsf{N}})$| is an essential parameter. For |$\kappa =0$|⁠, we recover |$\textbf{D}_1$| of (3.10). (b) Following our discussion and notations of Remark 3.28(b), we have also verified that the element of |${\mathcal C}_n$| corresponding under the anti-isomorphism |${\mathcal C}_n\to{\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{p}_{2n}})$| to |$\mathsf{H}(\hat{\epsilon }^{\pm },\hat{\mathsf{n}}^{\pm },\hat{c}^{\pm })$| with |$\epsilon ^{\pm }_{01}=1$|⁠, is conjugate to |$\hat{{\mathbb{H}}}^{\vec{k}}_2$| with |$k_1=0$| and |$k_{i+1}=\frac{\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}}{2}$| for any |$1\leq i\leq n-1$|⁠. (c) For completeness of our list (3.22, 3.29), let us present explicit formulas for the 1st hamiltonian |$\hat{\textbf{D}}_1$| of the quantum difference affine Toda systems (defined in [6]) for the remaining classical series |$D^{(1)}_n$| and |$B^{(1)}_n$| (as we could not find such formulas in the literature): In type |$D^{(1)}_n$|⁠, we have \begin{multline} \hat{\textbf{D}}_1=\sum_{i=1}^n (T_{2\varpi_i}+T_{-2\varpi_i})- (v-v^{-1})^2 \sum_{i=1}^{n-1} e^{-\varpi_i+\varpi_{i+1}} \left(T_{\varpi_i+\varpi_{i+1}}+T_{-\varpi_i-\varpi_{i+1}}\right)-\\[-.5pt] (v-v^{-1})^2 e^{-\varpi_{n-1}-\varpi_n}\left(T_{-\varpi_{n-1}+\varpi_n}+v^{-2} T_{\varpi_{n-1}-\varpi_n}\right)+ (v-v^{-1})^4 e^{-2\varpi_{n-1}}-\\[-.5pt] \kappa v^{-2n+2}(v-v^{-1})^2 e^{\varpi_1+\varpi_2}(T_{-\varpi_1+\varpi_2}+T_{\varpi_1-\varpi_2})+ \kappa v^{-2n+2}(v-v^{-1})^4 e^{2\varpi_2}. \end{multline} (3.30) In type |$B^{(1)}_n$|⁠, we have \begin{multline} \hat{\textbf{D}}_1=1+\sum_{i=1}^n (T_{2\varpi_i}+T_{-2\varpi_i})- (v^2-v^{-2})^2 \sum_{i=1}^{n-1} e^{-\varpi_i+\varpi_{i+1}} \left(T_{\varpi_i+\varpi_{i+1}}+T_{-\varpi_i-\varpi_{i+1}}\right)-\\[-.5pt] (v-v^{-1})^2 e^{-\varpi_n} \left(vT_{-\varpi_n}+v^{-1}T_{\varpi_n}\right)+v^{-2}(1-v^{-1})^2(v-v^{-1})^2 e^{-2\varpi_n}-\\[-.5pt] \kappa v^{-4n+2}(v^2-v^{-2})^2e^{\varpi_1+\varpi_2}(T_{-\varpi_1+\varpi_2}+T_{\varpi_1-\varpi_2})+ \kappa v^{-4n+2}(v^2-v^{-2})^4e^{2\varpi_2}. \end{multline} (3.31) For |$\kappa =0$|⁠, these formulas recover |$\textbf{D}_1$| of (3.13) and (3.15), respectively. 3.36 Explicit formulas and classification in type |$G_2$| Recall explicit formulas for the action of |$U_v(\mathfrak{g}_2)$| on its 1st fundamental representation |$V_1$|⁠. The space |$V_1$| has a basis |$\{w_i\}_{i=0}^6$|⁠, in which the action is given by the following formulas: \begin{align*} & E_1\colon w_0\mapsto (v+v^{-1})w_2,\ w_1\mapsto 0,\ w_2\mapsto 0,\ w_3\mapsto 0,\ w_4\mapsto w_3,\ w_5\mapsto (v+v^{-1})w_0,\\& \quad \ \ \ w_6\mapsto w_1,\\ & E_2\colon w_0\mapsto 0,\ w_1\mapsto 0,\ w_2\mapsto w_6,\ w_3\mapsto w_5,\ w_4\mapsto 0,\ w_5\mapsto 0,\ w_6\mapsto 0,\\ & F_1\colon w_0\mapsto w_5,\ w_1\mapsto w_6,\ w_2\mapsto w_0,\ w_3\mapsto w_4,\ w_4\mapsto 0,\ w_5\mapsto 0,\ w_6\mapsto 0,\\ & F_2\colon w_0\mapsto 0,\ w_1\mapsto 0,\ w_2\mapsto 0,\ w_3\mapsto 0,\ w_4\mapsto 0,\ w_5\mapsto w_3,\ w_6\mapsto w_2,\\& L_1\colon w_0\mapsto w_0, w_1\mapsto v^2w_1, w_2\mapsto vw_2, w_3\mapsto v^{-1}w_3, w_4\mapsto v^{-2}w_4, w_5\mapsto v^{-1}w_5,\\& \quad \ \ \ w_6\mapsto vw_6, \end{align*} \begin{align*} & L_2\colon w_0\mapsto w_0,\ w_1\mapsto v^3w_1,\ w_2\mapsto w_2,\ w_3\mapsto v^{-3}w_3,\ w_4\mapsto v^{-3}w_4,\ w_5\mapsto w_5,\\&\quad \ \ \ w_6\mapsto v^{3}w_6,\\ & K_1\colon w_0\mapsto w_0, w_1\mapsto vw_1, w_2\mapsto v^2w_2, w_3\mapsto vw_3, w_4\mapsto v^{-1}w_4, w_5\mapsto v^{-2}w_5,\\&\quad \ \ \, \, w_6\mapsto v^{-1}w_6,\\ & K_2\colon w_0\mapsto w_0,\ w_1\mapsto w_1,\ w_2\mapsto v^{-3}w_2,\ w_3\mapsto v^{-3}w_3,\ w_4\mapsto w_4,\ w_5\mapsto v^{3}w_5,\\&\quad \ \ \, \,w_6\mapsto v^{3}w_6. \end{align*} Let |$\varpi _1$| and |$\varpi _2$| be the weights of |$w_2$| and |$w_6$|⁠, respectively, so that |$(\varpi _1,\varpi _1)=(\varpi _2,\varpi _2)=2, (\varpi _1,\varpi _2)=-1$|⁠. Then the weights of |$w_0, w_1, w_3, w_4, w_5$| are equal to |$0, \varpi _1+\varpi _2, -\varpi _2, -\varpi _1-\varpi _2, -\varpi _1$|⁠, respectively. We also note that the simple roots are given by |$\alpha _1=\varpi _1, \alpha _2=-\varpi _1+\varpi _2$|⁠. Finally, we have |$\rho =5\alpha _1+3\alpha _2=2\varpi _1+3\varpi _2$| and |${\mathsf{d}}_1=1, {\mathsf{d}}_2=3$|⁠. To compute |$\textbf{D}_1$|⁠, we use the same strategy as for the classical types. Analogously to type |$B$|⁠, the operators |$E_1^2$| and |$F_1^2$| act nontrivially on |$V_1$|⁠, while |$\{E_i^r,F_i^r\}^{r>1+\delta _{i,1}}_{i=1,2}$| still act by zero on |$V_1$|⁠. Therefore, applying formula (3.1), we can replace |$R_{\alpha _2}$| by |$\bar{R}_{\alpha _2}:=1+(v^3-v^{-3})E_2\otimes F_2$| and |$R_{\alpha _1}$| by |$\bar{R}_{\alpha _1}:=1+(v-v^{-1})E_1\otimes F_1+cE_1^2\otimes F_1^2$| for |$c:=(1-v^{-1})(v-v^{-1})$|⁠. Let us now compute all the nonzero terms contributing to |$C^{\prime}_{V_1}$|⁠: Picking |$1$| out of each |$\bar{R}^{\textrm{op}}_i, \bar{R}_i$|⁠, we recover \begin{equation*} 1+v^2\cdot K_{2\varpi_1}+v^{-2}\cdot K_{-2\varpi_1}+v^8\cdot K_{2\varpi_2}+v^{-8}\cdot K_{-2\varpi_2}+ v^{10}\cdot K_{2\varpi_1+2\varpi_2}+v^{-10}\cdot K_{-2\varpi_1-2\varpi_2}. \end{equation*} Picking nontrivial terms only at |$\bar{R}_{\alpha _j}^{\textrm{op}}, \bar{R}_{\alpha _i}$|⁠, the result does not depend on |$\textrm{Or}^{\pm }$| (hence, the orderings |$\prec _{\pm }$|⁠) and the total contribution of the nonzero terms equals \begin{align*} & (v-v^{-1})^2 \left(v^2(v+v^{-1})F_1E_1K_{\varpi_1}+(v+v^{-1})F_1K_{-\varpi_1}E_1\right.\\ & \left.+\ v^{10}F_1K_{\varpi_2}E_1K_{\varpi_1+\varpi_2}+ v^{-8}F_1K_{-\varpi_1-\varpi_2}E_1K_{-\varpi_2}\right)\\ & \,+(v^3-v^{-3})^2 \left(v^8F_2K_{\varpi_1}E_2K_{\varpi_2}+v^{-2}F_2 K_{-\varpi_2}E_2K_{-\varpi_1}\right)\\& \,+c^2v^2(v+v^{-1})^2F_1^2 K_{-\varpi_1}E_1^2K_{\varpi_1}. \end{align*} The contribution of the remaining terms to |$C^{\prime}_{V_1}$| depends on |$\textrm{Or}^{\pm }$|⁠. Tracing back explicit formulas for the action of |$U_v(\mathfrak{g}_2)$| on |$V_1$|⁠, let us evaluate the total contribution of such terms for each of the four possible pairs |$(\textrm{Or}^+,\textrm{Or}^-)$|⁠.Case 1: if |$\textrm{Or}^{+}=\textrm{Or}^-$|⁠, then there are no other terms. Case 2: if |$\textrm{Or}^{+}\colon \alpha _1\leftarrow \alpha _2$| and |$\textrm{Or}^-\colon \alpha _1\to \alpha _2$|⁠, then the total contribution equals \begin{align*} & (v-v^{-1})^2(v^3-v^{-3})^2(v^8(v+v^{-1})F_2F_1E_1E_2K_{\varpi_2}+v^{-2}F_2F_1K_{-\varpi_1-\varpi_2}E_1E_2K_{-\varpi_1})+\\ & c^2(v^3-v^{-3})^2v^8(v+v^{-1})^2\cdot F_2F_1^2K_{-\varpi_1}E_1^2E_2K_{\varpi_2}. \end{align*} Case 3: if |$\textrm{Or}^{+}\colon \alpha _1\to \alpha _2$| and |$\textrm{Or}^-\colon \alpha _1\leftarrow \alpha _2$|⁠, then the total contribution equals \begin{align*} & (v-v^{-1})^2(v^3-v^{-3})^2(v^{10}F_1F_2K_{\varpi_1}E_2E_1K_{\varpi_1+\varpi_2}+(v+v^{-1})F_1F_2K_{-\varpi_2}E_2E_1)+\\ & c^2(v^3-v^{-3})^2v^2(v+v^{-1})^2\cdot F_1^2F_2K_{-\varpi_2}E_2E_1^2K_{\varpi_1}. \end{align*} Thus, we have listed all the nonzero terms contributing to |$C^{\prime}_{V_1}$|⁠. To obtain |$\tilde{\textbf{D}}_1=\bar{\textbf{D}}_{V_1}$|⁠, we should rewrite the above formulas via |$e_i, f_i$| and $$L_p=\begin{cases} K_{\varpi _1+\varpi _2}, & \ \textrm{if}\ p=1\\ K_{\varpi _1+2\varpi _2}, & \ \textrm{if}\ p=2 \end{cases},$$ moving all the Cartan terms to the middle, and then apply the characters |$\chi ^{\pm }$| with |$\chi ^+(e_i)=c^+_i, \chi ^-(f_i)=c^-_i$|⁠. Conjugating further by |$e^{\rho }$|⁠, we obtain the explicit formula for the 1st hamiltonian |$\textbf{D}_1$| of the type |$G_2$| modified quantum difference Toda system. To write it down, define constants |$\mathsf{m}_{i1}:=(\mathsf{n}^-_{i1}-\mathsf{n}^+_{i1})+(\mathsf{n}^-_{i2}-\mathsf{n}^+_{i2}),\ \mathsf{m}_{i2}:=(\mathsf{n}^-_{i1}-\mathsf{n}^+_{i1})+2(\mathsf{n}^-_{i2}-\mathsf{n}^+_{i2})$| and |$b_i:=(v_i-v_i^{-1})^2v_i^{\mathsf{n}^+_{ii}-\mathsf{n}^-_{ii}}c^+_ic^-_i$|⁠. Then we have \begin{multline} \textbf{D}_1=1+(T_{2\varpi_1}+T_{-2\varpi_1}+T_{2\varpi_2}+T_{-2\varpi_2}+T_{2\varpi_1+2\varpi_2}+T_{-2\varpi_1-2\varpi_2})+\\ b_1v^{-\mathsf{m}_{11}-4\mathsf{m}_{12}}\cdot e^{-\alpha_1} \left(v^{-1}(v+v^{-1})T_{(\mathsf{m}_{11}+1)\varpi_1+\mathsf{m}_{12}\varpi_2}+v(v+v^{-1})T_{(\mathsf{m}_{11}-1)\varpi_1+\mathsf{m}_{12}\varpi_2}\right.+ \nonumber \end{multline} (3.32) \begin{multline} \left.T_{(\mathsf{m}_{11}+1)\varpi_1+(\mathsf{m}_{12}+2)\varpi_2}+T_{(\mathsf{m}_{11}-1)\varpi_1+(\mathsf{m}_{12}-2)\varpi_2}\right)+\\ b_2v^{-\mathsf{m}_{21}-4\mathsf{m}_{22}}\cdot e^{-\alpha_2}\left(T_{(\mathsf{m}_{21}+1)\varpi_1+(\mathsf{m}_{22}+1)\varpi_2}+T_{(\mathsf{m}_{21}-1)\varpi_1+(\mathsf{m}_{22}-1)\varpi_2}\right)+\\ \frac{b_1^2(v+v^{-1})^2}{(1+v)^2}v^{-2+(\mathsf{n}^+_{11}-\mathsf{n}^-_{11})-(2\mathsf{m}_{11}+8\mathsf{m}_{12})}\cdot e^{-2\alpha_1}T_{2\mathsf{m}_{11}\varpi_1+2\mathsf{m}_{12}\varpi_2}+\\ \delta_{\epsilon^+_{12},-1}\delta_{\epsilon^-_{12},1}\cdot \left\{b_1b_2v^{-4+3(\mathsf{n}^+_{12}-\mathsf{n}^-_{12})-(\mathsf{m}_{11}+\mathsf{m}_{21}+4\mathsf{m}_{12}+4\mathsf{m}_{22})}\times\right. \\\left. e^{-\alpha_1-\alpha_2}\left((v+v^{-1})T_{(\mathsf{m}_{11}+\mathsf{m}_{21})\varpi_1+(\mathsf{m}_{12}+\mathsf{m}_{22}+1)\varpi_2}+vT_{(\mathsf{m}_{11}+\mathsf{m}_{21}-2)\varpi_1+(\mathsf{m}_{12}+\mathsf{m}_{22}-1)\varpi_2}\right)+\right. \\ \left. \frac{b_1^2b_2(v+v^{-1})^2}{(1+v)^2} v^{-8+(\mathsf{n}^+_{11}-\mathsf{n}^-_{11})+6(\mathsf{n}^+_{12}-\mathsf{n}^-_{12})-(2\mathsf{m}_{11}+\mathsf{m}_{21}+8\mathsf{m}_{12}+4\mathsf{m}_{22})}\times\right. \\ \left. e^{-2\alpha_1-\alpha_2} T_{(2\mathsf{m}_{11}+\mathsf{m}_{21}-1)\varpi_1+(2\mathsf{m}_{12}+\mathsf{m}_{22}+1)\varpi_2}\right\}+\\ \delta_{\epsilon^+_{12},1}\delta_{\epsilon^-_{12},-1}\cdot \left\{b_1b_2v^{3+3(\mathsf{n}^+_{12}-\mathsf{n}^-_{12})-(\mathsf{m}_{11}+\mathsf{m}_{21}+4\mathsf{m}_{12}+4\mathsf{m}_{22})}\times\right. \\ \left. e^{-\alpha_1-\alpha_2}\left(v(v+v^{-1})T_{(\mathsf{m}_{11}+\mathsf{m}_{21})\varpi_1+(\mathsf{m}_{12}+\mathsf{m}_{22}-1)\varpi_2}+T_{(\mathsf{m}_{11}+\mathsf{m}_{21}+2)\varpi_1+(\mathsf{m}_{12}+\mathsf{m}_{22}+1)\varpi_2}\right)+\right. \\ \left. \frac{b_1^2b_2(v+v^{-1})^2}{(1+v)^2} v^{4+(\mathsf{n}^+_{11}-\mathsf{n}^-_{11})+6(\mathsf{n}^+_{12}-\mathsf{n}^-_{12})-(2\mathsf{m}_{11}+\mathsf{m}_{21}+8\mathsf{m}_{12}+4\mathsf{m}_{22})}\times\right. \\ \left. e^{-2\alpha_1-\alpha_2} T_{(2\mathsf{m}_{11}+\mathsf{m}_{21}+1)\varpi_1+(2\mathsf{m}_{12}+\mathsf{m}_{22}-1)\varpi_2}\right\}. \end{multline} (3.32) Remark 3.37. If |$\epsilon ^+=\epsilon ^-$|⁠, then the terms with |$\delta $|’s are vacuous. If we further set |$\mathsf{n}^+=\mathsf{n}^-$| and |$c^{\pm }_i=\pm 1$| for all |$i$|⁠, then we obtain the formula for the 1st hamiltonian of the type |$G_2$| quantum difference Toda lattice as defined in [6]: \begin{multline} \textbf{D}_1=1+(T_{2\varpi_1}+T_{-2\varpi_1}+T_{2\varpi_2}+T_{-2\varpi_2}+T_{2\varpi_1+2\varpi_2}+T_{-2\varpi_1-2\varpi_2})-\\ (v-v^{-1})^2\cdot e^{-\alpha_1}(v^{-1}(v+v^{-1})T_{\varpi_1}+v(v+v^{-1})T_{-\varpi_1}+T_{\varpi_1+2\varpi_2}+T_{-\varpi_1-2\varpi_2})-\\ (v^3-v^{-3})^2\cdot e^{-\alpha_2}(T_{\varpi_1+\varpi_2}+T_{-\varpi_1-\varpi_2})+ v^{-2}(1-v^{-1})^2(v^2-v^{-2})^2\cdot e^{-2\alpha_1}. \end{multline} (3.33) Let |${\mathcal{G}}_2$| be the associative |${\mathbb{C}}(v)$|-algebra generated by |$\{{\mathsf{w}}_i^{\pm 1}, {\mathsf{D}}_i^{\pm 1}\}_{i=1}^2$| subject to \begin{align*} & [{\mathsf{w}}_1,{\mathsf{w}}_2]=[{\mathsf{D}}_1,{\mathsf{D}}_2]=0,\ {\mathsf{w}}_i^{\pm 1}{\mathsf{w}}_i^{\mp 1}={\mathsf{D}}_i^{\pm1}{\mathsf{D}}_i^{\mp 1}=1,\\ & {\mathsf{D}}_1{\mathsf{w}}_1=v^2{\mathsf{w}}_1{\mathsf{D}}_1,\ {\mathsf{D}}_1{\mathsf{w}}_2=v^{-1}{\mathsf{w}}_2{\mathsf{D}}_1,\ {\mathsf{D}}_2{\mathsf{w}}_1=v^{-3}{\mathsf{w}}_1{\mathsf{D}}_2,\ {\mathsf{D}}_2{\mathsf{w}}_2=v^3{\mathsf{w}}_2{\mathsf{D}}_2. \end{align*} Consider the anti-isomorphism from |${\mathcal{G}}_2$| to the algebra |${\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{g}_2})$| of Section 2.5, sending |${\mathsf{w}}_i\mapsto T_{-\varpi _i}, {\mathsf{D}}_i\mapsto e^{-\alpha _i}$|⁠. Let |$\mathsf{H}=\mathsf{H}(\epsilon ^{\pm }, \mathsf{n}^{\pm }, c^{\pm })$| be the element of |${\mathcal{G}}_2$| that corresponds to |$\textbf{D}_1$| under this anti-isomorphism. The following is the key property of |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| in type |$G_2$|⁠. Proposition 3.38. |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| depends only on |$\epsilon :=\frac{\epsilon ^+_{12}-\epsilon ^-_{12}}{2}\in \{-1,0,1\}$|⁠, up to algebra automorphisms of |${\mathcal{G}}_2$|⁠. The proof of Proposition 3.38 is completely analogous to that of Proposition 3.11 given in Appendix A; we leave the details to the interested reader. Proposition 3.38 implies that given two pairs of Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| and |$(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| with |$\epsilon ^+_{12}-\epsilon ^-_{12}=\tilde{\epsilon }^+_{12}-\tilde{\epsilon }^-_{12}$|⁠, there exists an algebra automorphism of |${\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{g}_2})$| that maps the 1st hamiltonian |$\textbf{D}_1(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |$\textbf{D}_1(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. As we will see in Appendix B, the same automorphism maps the modified quantum Toda system |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |${\mathcal{T}}(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. Remark 3.39. For completeness of our list (3.22, 3.29, 3.30, 3.31), let us present the explicit formula for the 1st hamiltonian |$\hat{\textbf{D}}_1$| of the type |$G^{(1)}_2$| quantum difference affine Toda system: \begin{multline} \hat{\textbf{D}}_1=1+(T_{2\varpi_1}+T_{-2\varpi_1}+T_{2\varpi_2}+T_{-2\varpi_2}+T_{2\varpi_1+2\varpi_2}+T_{-2\varpi_1-2\varpi_2})-\\ (v-v^{-1})^2 e^{-\varpi_1}(v^{-1}(v+v^{-1})T_{\varpi_1}+v(v+v^{-1})T_{-\varpi_1}+T_{\varpi_1+2\varpi_2}+T_{-\varpi_1-2\varpi_2})-\\ (v^3-v^{-3})^2 e^{\varpi_1-\varpi_2}(T_{\varpi_1+\varpi_2}+T_{-\varpi_1-\varpi_2})+ v^{-2}(1-v^{-1})^2(v^2-v^{-2})^2\cdot e^{-2\varpi_1}-\\ \kappa v^{-12}(v^3-v^{-3})^2 e^{\varpi_1+2\varpi_2}(T_{\varpi_1}+T_{-\varpi_1})+ \kappa v^{-12}(v-v^{-1})^2(v^3-v^{-3})^2 e^{2\varpi_2}. \end{multline} (3.34) For |$\kappa =0$|⁠, this recovers |$\textbf{D}_1$| of (3.33). 4 Whittaker Vectors and Their Pairing In this section, we study a pairing of two general Whittaker vectors (associated with a pair of Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$|⁠) in universal Verma modules, following [7]. We obtain a fermionic formula for the corresponding terms |$\tilde{J}_{\beta }$|⁠. We show that their generating series is a natural solution of the modified quantum difference Toda system |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| of Section 2. This provides a natural generalization of [7, Section 3], where |$\epsilon ^+=\epsilon ^-$| and |$\mathsf{n}^+=\mathsf{n}^-$|⁠. 4.1 Whittaker vectors Following the notations of [7], consider |$U_v(\mathfrak{g})$| and |$U_{v^{-1}}(\mathfrak{g})$|⁠, whose generators will be denoted by |$E_i,F_i,L_i$| and |$\bar{E}_i,\bar{F}_i,\bar{L}_i$|⁠, respectively. In contrast to [7], we will work with universal Verma modules. Let |$\{u_i\}_{i=1}^n$| be indeterminates and consider an extension |${\mathsf{k}}:={\mathbb{C}}(v^{1/\mathsf{N}},u_1,\ldots ,u_n)$| of |${\mathbb{C}}(v^{1/\mathsf{N}})$|⁠. Let |$U_v(\mathfrak{g})^{\leq }$| be the subalgebra of |$U_v(\mathfrak{g})$| generated by |$\{L_i^{\pm 1},F_i\}_{i=1}^n$| and consider its action on |${\mathsf{k}}$| with |$F_i$| acting trivially and |$L_i$| acting via multiplication by |$u_i$|⁠. We define the universal Verma module |${\mathcal{V}}$| over |$U_v(\mathfrak{g})$| as |${\mathcal{V}}:=U_v(\mathfrak{g})\otimes _{U_v(\mathfrak{g})^{\leq }} {\mathsf{k}}$|⁠. It is generated by |$\textbf{1}\in{\mathsf{k}}$| such that |$E_i(\textbf{1})=0$| and |$L_i(\textbf{1})=u_i\cdot \textbf{1}$| for |$1\leq i\leq n$|⁠. We define the formal symbol |$\lambda :=\sum _{i=1}^n \frac{\log (u_i)}{{\mathsf{d}}_i\log (v)}\alpha _i$|⁠, which will appear only as an index or in the context of the homomorphism |$v^{(\lambda ,\cdot )}\colon P\to{\mathsf{k}}$| defined by |$P\ni m_i\omega _i\mapsto v^{(\lambda ,\sum _{i=1}^n m_i\omega _i)}=\prod _{i=1}^n u_i^{m_i}$|⁠, so that |$K_{\mu }(\textbf{1})=v^{(\lambda ,\mu )}\cdot \textbf{1}$| for |$\mu \in P$|⁠. In particular, |${\mathcal{V}}$| is graded by |$Q_+$|⁠: |${\mathcal{V}}=\oplus _{\beta \in Q_+} {\mathcal{V}}_{\beta }$| with |${\mathcal{V}}_{\beta }=\{w\in{\mathcal{V}}| K_{\mu }(w)=v^{(\mu ,\lambda -\beta )}w\ (\mu \in P)\}$|⁠. Similarly, let |$\bar{{\mathcal{V}}}$| be the universal Verma module over |$U_{v^{-1}}(\mathfrak{g})$| generated by the highest weight vector |$\bar{\textbf{1}}$| such that |$\bar{E}_i(\bar{\textbf{1}})=0$| and |$\bar{L}_i(\bar{\textbf{1}})=u_i^{-1}\cdot \bar{\textbf{1}}$| for |$1\leq i\leq n$|⁠. It is also |$Q_+$|-graded: |$\bar{{\mathcal{V}}}=\oplus _{\beta \in Q_+} \bar{{\mathcal{V}}}_{\beta }$| with |$\bar{{\mathcal{V}}}_{\beta }=\{w\in \bar{{\mathcal{V}}}| \bar{K}_{\mu }(w)=v^{-(\mu ,\lambda -\beta )}w\ (\mu \in P)\}$|⁠. Remark 4.2. One can alternatively work with the standard Verma modules |${\mathcal{V}}^{\lambda }$| and |$\bar{{\mathcal{V}}}^{\lambda }, \lambda \in P$| (one should further require |$\lambda $| to be strictly antidominant for the existence of Whittaker vectors), so that |$u_i=v^{(\lambda ,\omega _i)}\in{\mathbb{C}}(v^{1/\mathsf{N}})$|⁠. This viewpoint is used in [7]. We prefer the current exposition as it is compatible with our discussion in Section 5. Nevertheless, motivated by the above standard set-up, we will freely use the above notation |$v^{(\lambda ,\mu )}$| for |$\mu \in P$|⁠. There is a unique nondegenerate |${\mathsf{k}}$|-bilinear pairing |$(\cdot ,\cdot )\colon{\mathcal{V}}\times \bar{{\mathcal{V}}}\to{\mathsf{k}}$| such that |$(\textbf{1}, \bar{\textbf{1}})=1$| and |$(xw,w^{\prime})=(w,\sigma (x)w^{\prime})$| for all |$x\in U_v(\mathfrak{g}), w\in{\mathcal{V}}, w^{\prime}\in \bar{{\mathcal{V}}}$|⁠, where the algebra anti-isomorphism |$\sigma \colon U_v(\mathfrak{g})\to U_{v^{-1}}(\mathfrak{g})$| is determined by |$\sigma (E_i)=\bar{F}_i, \sigma (F_i)=\bar{E}_i, \sigma (L_i)=\bar{L}_i^{-1}$|⁠. Remark 4.3. One can alternatively work with a single universal Verma module |${\mathcal{V}}$| over |$U_v(\mathfrak{g})$| endowed with the Shapovalov form |$(\cdot ,\cdot )\colon{\mathcal{V}}\times{\mathcal{V}}\to{\mathsf{k}}$|⁠; see our discussion in Remark 5.12. For the key definition of this section, consider the completions |${\mathcal{V}}^{\wedge }, \bar{{\mathcal{V}}}^{\wedge }$| of |${\mathcal{V}}, \bar{{\mathcal{V}}}$|⁠, defined via \begin{equation*} {\mathcal{V}}^{\wedge}:=\prod_{\beta\in Q_+} {\mathcal{V}}_{\beta},\ \bar{{\mathcal{V}}}^{\wedge}:=\prod_{\beta\in Q_+} \bar{{\mathcal{V}}}_{\beta}. \end{equation*} Given a pair of Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| (now |$c^{\pm }_i\in{\mathsf{k}}^{\times }$|⁠), define |$\nu ^{\pm }_i\in P$| via |$\nu ^{\pm }_i:=\sum _{k=1}^n \mathsf{n}_{ik}^{\pm }\omega _k$|⁠, so that |$(\nu ^{\pm }_i,\alpha _j)-(\nu ^{\pm }_j,\alpha _i)=\epsilon ^{\pm }_{i,j}b_{i,j}$|⁠. We have associated Whittaker vectors \begin{equation*} \theta=\theta(\epsilon^+,\mathsf{n}^+,c^+)=\sum_{\beta\in Q_+} \theta_{\beta}\in{\mathcal{V}}^{\wedge} \ \ (\textrm{with}\ \theta_{\beta}\in{\mathcal{V}}_{\beta}) \end{equation*} and \begin{equation*} \bar{\theta}=\bar{\theta}(\epsilon^-,\mathsf{n}^-,c^-)=\sum_{\beta\in Q_+} \bar{\theta}_{\beta}\in \bar{{\mathcal{V}}}^{\wedge} \ \ (\textrm{with}\ \bar{\theta}_{\beta}\in \bar{{\mathcal{V}}}_{\beta}), \end{equation*} which are uniquely determined by the following conditions: \begin{equation} \theta_0=\textbf{1},\ E_iK_{\nu^+_i}(\theta)=c^+_i\cdot \theta\ \ \textrm{and}\ \ \bar{\theta}_0=\bar{\textbf{1}},\ \bar{E}_i\bar{K}_{\nu^-_i}(\bar{\theta})=c^-_i\cdot \bar{\theta}. \end{equation} (4.1) Remark 4.4. This is a direct generalization of the classical notion of Whittaker vectors for Lie algebras as defined by B. Kostant in his milestone work on the subject [14]. 4.5 Pairing of Whittaker vectors Set |$(t;t)_r:=\prod _{k=1}^r (1-t^k)$| for |$r\in{\mathbb{Z}}_{>0}$|⁠, and |$(t;t)_0:=1$|⁠. Choose convex orderings |$\prec _{\pm }$| on |$\Delta _+$| such that |$\epsilon ^{\pm }_{ij}=-1\Rightarrow \alpha _i\prec _{\pm } \alpha _j$|⁠, cf. Section 3.4. Let |$\alpha _{i^{\pm }_1}\prec _{\pm }\cdots \prec _{\pm }\alpha _{i^{\pm }_n}$| be the simple roots ordered with respect to |$\prec _{\pm }$|⁠. For |$1\leq i\ne j\leq n$| we write |$i\prec _{\pm } j$| if |$\alpha _i\prec _{\pm } \alpha _j$|⁠. Define \begin{equation} J_{\beta}=J_{\beta}(\epsilon^{\pm},\mathsf{n}^{\pm},c^{\pm}):= \left(\theta_{\beta}(\epsilon^+,\mathsf{n}^+,c^+),\bar{\theta}_{\beta}(\epsilon^-,\mathsf{n}^-,c^-)\right). \end{equation} (4.2) Following [7, (3.11)] (we note that |$\tilde{J}_{\beta }$| is denoted by |$J^{\lambda }_{\beta }$| in [7]) we also consider its slight modification \begin{equation} \tilde{J}_{\beta}=\tilde{J}_{\beta}(\epsilon^{\pm},\mathsf{n}^{\pm},c^{\pm}):= v^{-(\beta,\beta)/2+(\lambda,\beta)} \left(\theta_{\beta}(\epsilon^+,\mathsf{n}^+,c^+),\bar{\theta}_{\beta}(\epsilon^-,\mathsf{n}^-,c^-)\right). \end{equation} (4.3) For |$\beta \notin Q_+$|⁠, we set |$J_{\beta }:=0$| and |$\tilde{J}_{\beta }:=0$|⁠. Our 1st result provides a recursive formula for |$\tilde{J}_{\beta }$|⁠. Theorem 4.6. We have \begin{equation} \tilde{J}_{\beta}=\sum_{0\leq \gamma\leq \beta} \frac{1}{(v^2)_{\beta-\gamma}} v^{(\gamma,\gamma)-2(\lambda+\rho,\gamma)}c^{\beta-\gamma}v^{\tau_{\lambda}(\beta-\gamma,\beta)} \cdot \tilde{J}_{\gamma}, \end{equation} (4.4) where |$(v^2)_{\alpha }:=\prod _{i=1}^n (v_i^2;v_i^2)_{m_i}, c^{\alpha }:=\prod _{i=1}^n (-c^+_ic^-_i(v_i-v_i^{-1})^2)^{m_i}, \tau _{\lambda }(\alpha ,\beta ):=\sum _{i=1}^n m_i(\nu ^-_i-\nu ^+_i,\lambda -\beta )+ \sum _{j\prec _+ i} m_im_j(\nu ^-_i-\nu ^+_i,\alpha _j)+ \sum _{i=1}^n \frac{m_i(m_i-1)}{2}(\nu ^-_i-\nu ^+_i,\alpha _i)$| for |$\alpha =\sum _{i=1}^n m_i\alpha _i\in Q_+$|⁠. Proof. The proof is completely analogous to that of [7, Theorem 3.1] and is based on an evaluation of |$(C(\theta _{\beta }),\bar{\theta }_{\beta })$| in two different ways, where |$C$| is the Drinfeld Casimir element. Solving the recursive relation (4.4), one obtains an explicit fermionic formula for |$\tilde{J}_{\beta }$|⁠. Theorem 4.7. We have \begin{equation} \tilde{J}_{\beta}=c^{\beta}\cdot \sum_{\underline{\beta}=\{\beta^{(t)}\}_{t=0}^{\infty}\in Q_+^{\mathbb{N}}}^{\sum_{t=0}^{\infty} \beta^{(t)}=\beta} \frac{v^{2B(\underline{\beta})+R(\underline{\beta})}}{\prod_{t=0}^{\infty} (v^2)_{\beta^{(t)}}}, \end{equation} (4.5) where we set |$R(\underline{\beta }):=\sum _{t=0}^{\infty } \tau _{\lambda }\left (\beta ^{(t)},\beta -\sum _{s=0}^{t-1}\beta ^{(s)}\right )$| and \begin{equation*} B(\underline{\beta}):=\frac{1}{2}\sum_{t,t^{\prime}=0}^{\infty} \min(t,t^{\prime})\left(\beta^{(t)},\beta^{(t^{\prime})}\right)- \sum_{t=0}^{\infty} t \left(\lambda+\rho,\beta^{(t)}\right). \end{equation*} Proof. We will give a direct proof as the general machinery of fermionic formulas developed in [7, Section 2] does not apply to our set-up. The formula is obvious for |$\beta =0$|⁠. From now on, fix |$\beta>0$| (that is, |$\beta \in Q_+\backslash \{0\}$|⁠). Let us rewrite the equality (4.4) as \begin{equation*} \left(1-v^{(\beta,\beta)-2(\lambda+\rho,\beta)}\right)\tilde{J}_{\beta}= \sum_{0<\beta_1\leq \beta}\frac{1}{(v^2)_{\beta_1}}v^{(\beta-\beta_1,\beta-\beta_1)-2(\lambda+\rho,\beta-\beta_1)} c^{\beta_1}v^{\tau_{\lambda}(\beta_1,\beta)}\tilde{J}_{\beta-\beta_1}. \end{equation*} We apply the same formula for |$\tilde{J}_{\beta -\beta _1}$| if |$\beta _1<\beta $|⁠. Proceeding in the same way, we finally obtain \begin{equation} \tilde{J}_{\beta}=\sum_{d\geq 1}\sum_{\beta_1,\ldots,\beta_d>0}^{\beta_1+\ldots+\beta_d=\beta} \frac{\prod_{e=1}^d c^{\beta_e}v^{\tau_{\lambda}(\beta_e,\beta_e+\ldots+\beta_{d})} v^{(\beta_{e+1}+\ldots+\beta_d,\beta_{e+1}+\ldots+\beta_d)-2(\lambda+\rho,\beta_{e+1}+\ldots+\beta_d)}} {\prod_{e=1}^d (v^2)_{\beta_e} (1-v^{(\beta_e+\ldots+\beta_d, \beta_e+\ldots+\beta_d)-2(\lambda+\rho,\beta_e+\ldots+\beta_d)})}. \end{equation} (4.6) On the other hand, the summation in the right-hand side of (4.5) is over all |$\underline{\beta }=\{\beta ^{(t)}\}_{t=0}^{\infty }\in Q_+^{\mathbb{N}}$| with |$\sum _{t=0}^{\infty }\beta ^{(t)}=\beta $|⁠. Such sequences are in bijection with tuples |$\{d,\{\beta _e\}_{e=1}^d,\{t_e\}_{e=1}^d|d\geq 1, \beta _e>0, t_e\in{\mathbb{N}}, \sum _{e=1}^d\beta _e=\beta \}$| via |$\beta ^{(t_1+\ldots +t_e+e-1)}=\beta _e\ (1\leq e\leq d)$| and |$\beta ^{(t)}=0$| otherwise. Hence, the right-hand side of (4.5) equals \begin{multline} \sum_{d\geq 1}\sum_{\beta_1,\ldots,\beta_d>0}^{\beta_1+\ldots+\beta_d=\beta} \left\{\frac{c^{\beta} v^{\sum_{e=1}^d \tau_{\lambda}(\beta_e,\beta-\beta_1-\ldots-\beta_{e-1})}}{\prod_{e=1}^d (v^2)_{\beta_e}} \times\right.\\ \left.\sum_{t_1,\ldots,t_d\geq 0} v^{\sum_{e=1}^d (t_1+\ldots+t_e+e-1)(\beta_e,\beta_e)+ 2\sum_{e<e^{\prime}}(t_1+\ldots+t_e+e-1)(\beta_e,\beta_{e^{\prime}})-2(\lambda+\rho,\sum_{e=1}^d (t_1+\ldots+t_e+e-1)\beta_e)}\right\}. \end{multline} (4.7) It is straightforward to verify that the right-hand side of (4.6) coincides with (4.7). 4.8 |$J$|-functions: eigenfunctions of modified quantum difference Toda systems Recall the elements |$\omega _i^{\vee }=\omega _i/{\mathsf{d}}_i\in P\otimes _{{\mathbb{Z}}}{\mathbb{Q}}$| satisfying |$(\alpha _j,\omega _i^{\vee })=\delta _{i,j}$|⁠. Consider a vector space |$N_{\lambda }$| that consists of all formal sums |$\left \{\sum _{\beta \in Q} a_{\beta } \underline{y}^{\beta -\lambda }|a_{\beta }\in{\mathsf{k}}\right \}$| for which there exists |$\beta _0\in Q$| such that |$a_{\beta }=0$| unless |$\beta -\beta _0\in Q_+$|⁠, where |$\underline{y}^{\beta -\lambda }$| is used to denote |$\prod _{i=1}^n y_i^{(\beta -\lambda ,\omega _i^{\vee })}$|⁠. The vector space |$N_{\lambda +\rho }$| is defined analogously with |$\lambda $| being replaced by |$\lambda +\rho $|⁠. Consider the natural action of the algebra |${\mathcal{D}}_v(H^{\textrm{ad}})$| of Section 2.5 on the vector space |$N_{\lambda }$|⁠, determined by \begin{equation} T_{\mu}(\underline{y}^{\beta-\lambda})=v^{-(\mu,\beta-\lambda)}\underline{y}^{\beta-\lambda},\ e^{\alpha}(\underline{y}^{\beta-\lambda})=\underline{y}^{\beta-\alpha-\lambda}\ \textrm{for}\ \alpha,\beta\in Q,\mu\in P. \end{equation} (4.8) The action of |${\mathcal{D}}_v(H^{\textrm{ad}})$| on |$N_{\lambda +\rho }$| is defined analogously. Consider the following generating functions of the terms |$J_{\beta }$| defined in (4.2): \begin{equation} \begin{split} & \tilde{J}=\tilde{J}(\{y_i\}_{i=1}^n):= \sum_{\beta\in Q_+} J_{\beta} \prod_{i=1}^n y_i^{(\beta-\lambda,\omega_i^{\vee})}\in N_{\lambda},\\ & J=J(\{y_i\}_{i=1}^n):= \sum_{\beta\in Q_+} J_{\beta} \prod_{i=1}^n y_i^{(\beta-\lambda-\rho,\omega_i^{\vee})}\in N_{\lambda+\rho}. \end{split} \end{equation} (4.9) Recall the difference operators |$\tilde{\textbf{D}}_V,\textbf{D}_V\in{\mathcal{D}}_v(H^{\textrm{ad}})$| of Section 2.5, associated with the pair of Sevostyanov triples |$(\epsilon ^{\pm }, \mathsf{n}^{\pm }, c^{\pm })$| and a finite-dimensional |$U_v(\mathfrak{g})$|-representation |$V$|⁠. The following is the key result of this section. Theorem 4.9. (a) We have |$\tilde{\textbf{D}}_V(\tilde{J})=\textrm{tr}_V\left (v^{2(\lambda +\rho )}\right )\cdot \tilde{J}$|⁠. (b) We have |$\textbf{D}_V(J)=\textrm{tr}_V\left (v^{2(\lambda +\rho )}\right )\cdot J$|⁠. Proof. First, we note that part (a) implies part (b), due to |$\textbf{D}_V=e^{\rho } \tilde{\textbf{D}}_V e^{-\rho }$| and |$J=e^{\rho }(\tilde{J})$|⁠. The proof of part (a) is based on an evaluation of |$\left (C_V(\theta _{\beta }),\bar{\theta }_{\beta }\right )$| in two different ways, where |$C_V$| is the central element of (2.2). On the one hand, |$C_V$| acts on |${\mathcal{V}}$| as a multiplication by |$\textrm{tr}_V(v^{2(\lambda +\rho )})$| (since |$C_V$| is central, |${\mathcal{V}}$| is generated by |$\textbf{1}$|⁠, and |$C_V(\textbf{1})=\textrm{tr}_V(v^{2(\lambda +\rho )})\cdot \textbf{1}$|⁠), so that |$\left (C_V(\theta _{\beta }),\bar{\theta }_{\beta }\right )=\textrm{tr}_V(v^{2(\lambda +\rho )})\cdot J_{\beta }$|⁠. On the other hand, we can use the explicit formula for |$C_V$|⁠. Let |$\{w_k\}_{k=1}^N$| be a weight basis of |$V$|⁠, and |$\mu _k\in P$| be the weight of |$w_k$|⁠, cf. Section 2.5. Then we have \begin{multline} C_V(\theta_{\beta})=\sum_{\vec{m}=(m_1,\ldots,m_n)\in{\mathbb{N}}^n}^{1\leq k\leq N} c(\vec{m})\cdot v^{(\lambda-\beta+2\rho,\mu_k)+(\lambda-\beta+\alpha(\vec{m}),\mu_k-\alpha(\vec{m}))}\times\\ \left\langle w_k\middle| E_{\alpha_{i^-_1}}^{m_{i^-_1}}\cdots E_{\alpha_{i^-_n}}^{m_{i^-_n}}\cdot F_{\alpha_{i^+_1}}^{m_{i^+_1}}\cdots F_{\alpha_{i^+_n}}^{m_{i^+_n}}\middle| w_k\right\rangle\cdot F_{\alpha_{i^-_1}}^{m_{i^-_1}}\cdots F_{\alpha_{i^-_n}}^{m_{i^-_n}}\cdot E_{\alpha_{i^+_1}}^{m_{i^+_1}}\cdots E_{\alpha_{i^+_n}}^{m_{i^+_n}}(\theta_{\beta}), \end{multline} (4.10) where |$\langle w_k| x | w_k\rangle $| is the matrix coefficient of |$x\in U_v(\mathfrak{g})$|⁠, |$\alpha (\vec{m}):=\sum _{i=1}^n m_i\alpha _i\in Q_+$|⁠, and |$c(\vec{m})\in{\mathbb{C}}(v)$| are certain coefficients for which we currently do not need explicit formulas. Using the defining property of |$(\cdot ,\cdot )$|⁠, we get \begin{equation*} \left(F_{\alpha_{i^-_1}}^{m_{i^-_1}}\cdots F_{\alpha_{i^-_n}}^{m_{i^-_n}}\cdot E_{\alpha_{i^+_1}}^{m_{i^+_1}}\cdots E_{\alpha_{i^+_n}}^{m_{i^+_n}}(\theta_{\beta}),\bar{\theta}_{\beta}\right)= \left(E_{\alpha_{i^+_1}}^{m_{i^+_1}}\cdots E_{\alpha_{i^+_n}}^{m_{i^+_n}}(\theta_{\beta}), \bar{E}_{\alpha_{i^-_n}}^{m_{i^-_n}}\cdots \bar{E}_{\alpha_{i^-_1}}^{m_{i^-_1}}(\bar{\theta}_{\beta})\right). \end{equation*} To evaluate the pairing in the right-hand side, note that the defining conditions (4.1) of the Whittaker vectors imply |$E_iK_{\nu ^+_i}(\theta _{\gamma })=c^+_i\theta _{\gamma -\alpha _i}$| and |$\bar{E}_i\bar{K}_{\nu ^-_i}(\bar{\theta }_{\gamma })=c^-_i\bar{\theta }_{\gamma -\alpha _i}$| for |$\gamma \in Q$|⁠; hence, |$E_i(\theta _{\gamma })=c^+_iv^{-(\nu ^+_i,\lambda -\gamma )}\theta _{\gamma -\alpha _i}$| and |$\bar{E}_i(\bar{\theta }_{\gamma })=c^-_iv^{(\nu ^-_i,\lambda -\gamma )}\bar{\theta }_{\gamma -\alpha _i}$|⁠. Applying this iteratively, we find \begin{align*} & E_{\alpha_{i^+_1}}^{m_{i^+_1}}\cdots E_{\alpha_{i^+_n}}^{m_{i^+_n}}(\theta_{\beta})= \prod_{i=1}^n (c^+_i)^{m_i} v^{\tau^+_{\lambda}(\vec{m},\beta)}\cdot \theta_{\beta-\alpha(\vec{m})},\\ & \bar{E}_{\alpha_{i^-_n}}^{m_{i^-_n}}\cdots \bar{E}_{\alpha_{i^-_1}}^{m_{i^-_1}}(\bar{\theta}_{\beta})= \prod_{i=1}^n (c^-_i)^{m_i} v^{\tau^-_{\lambda}(\vec{m},\beta)}\cdot \bar{\theta}_{\beta-\alpha(\vec{m})} \end{align*} with |$\tau ^{\pm }_{\lambda }(\vec{m},\beta )$| given by |$\tau ^+_{\lambda }(\vec{m},\beta ):=-\sum _{k=1}^n \sum _{r=0}^{m_{i^+_k}-1} \left (\nu ^+_{i^+_k},\lambda -\beta +r\alpha _{i^+_k}+\sum _{s=k+1}^n m_{i^+_{s}}\alpha _{i^+_{s}}\right )$| and |$\tau ^-_{\lambda }(\vec{m},\beta ):=\sum _{k=1}^n \sum _{r=0}^{m_{i^-_k}-1} \left (\nu ^-_{i^-_k},\lambda -\beta +r\alpha _{i^-_k}+\sum _{s=1}^{k-1} m_{i^-_{s}}\alpha _{i^-_{s}}\right )$|⁠. Summarizing all these calculations, we obtain the following equality: \begin{multline} \sum_{\vec{m}=(m_1,\ldots,m_n)\in{\mathbb{N}}^n}^{1\leq k\leq N} c(\vec{m})\prod_{i=1}^n (c^+_ic^-_i)^{m_i}\cdot v^{(\lambda-\beta+2\rho,\mu_k)+(\lambda-\beta+\alpha(\vec{m}),\mu_k-\alpha(\vec{m}))+ \tau^+_{\lambda}(\vec{m},\beta)+\tau^-_{\lambda}(\vec{m},\beta)}\times\\ \left\langle w_k\middle| E_{\alpha_{i^-_1}}^{m_{i^-_1}}\cdots E_{\alpha_{i^-_n}}^{m_{i^-_n}}\cdot F_{\alpha_{i^+_1}}^{m_{i^+_1}}\cdots F_{\alpha_{i^+_n}}^{m_{i^+_n}}\middle| w_k\right\rangle\cdot J_{\beta-\alpha(\vec{m})}=\textrm{tr}_V\left(v^{2(\lambda+\rho)}\right)\cdot J_{\beta}. \end{multline} (4.11) Let us now compute |$\tilde{\textbf{D}}_V(\tilde{J})$|⁠. First, we need to rewrite |$F_{\alpha _{i^-_1}}^{m_{i^-_1}}\cdots F_{\alpha _{i^-_n}}^{m_{i^-_n}}$| and |$E_{\alpha _{i^+_1}}^{m_{i^+_1}}\cdots E_{\alpha _{i^+_n}}^{m_{i^+_n}}$| in terms of the Sevostyanov generators |$e_i,f_i$| and Cartan terms, moving the latter to the right of |$f_i$|’s and to the left of |$e_i$|’s. We have \begin{align*} & F_{\alpha_{i^-_1}}^{m_{i^-_1}}\cdots F_{\alpha_{i^-_n}}^{m_{i^-_n}}= v^{\tilde{\tau}^-_{\lambda}(\vec{m})} f_{\alpha_{i^-_1}}^{m_{i^-_1}}\cdots f_{\alpha_{i^-_n}}^{m_{i^-_n}}\cdot K_{\sum_{i=1}^n m_i\nu^-_i}, \\ & E_{\alpha_{i^+_1}}^{m_{i^+_1}}\cdots E_{\alpha_{i^+_n}}^{m_{i^+_n}}= v^{\tilde{\tau}^+_{\lambda}(\vec{m})} K_{-\sum_{i=1}^n m_i\nu^+_i}\cdot e_{\alpha_{i^+_1}}^{m_{i^+_1}}\cdots e_{\alpha_{i^+_n}}^{m_{i^+_n}}, \end{align*} where |$\tilde{\tau }^{\pm }_{\lambda }(\vec{m})$| are given by |$\tilde{\tau }^+_{\lambda }(\vec{m}):=\sum _{k=1}^n \sum _{r=1}^{m_{i^+_k}} \left (\nu ^+_{i^+_k},r\alpha _{i^+_k}+\sum _{s=1}^{k-1} m_{i^+_s}\alpha _{i^+_{s}}\right )$| and |$\tilde{\tau }^-_{\lambda }(\vec{m}):=-\sum _{k=1}^n \sum _{r=1}^{m_{i^-_k}} \left (\nu ^-_{i^-_k},r\alpha _{i^-_k}+\sum _{s=k+1}^n m_{i^-_s}\alpha _{i^-_{s}}\right )$|⁠. Tracing back the definition of |$\tilde{\textbf{D}}_V$|⁠, we find \begin{multline} \tilde{\textbf{D}}_V(\tilde{J})=\tilde{\textbf{D}}_V\left(\sum_{\tilde{\beta}\in Q_+} J_{\tilde{\beta}}\underline{y}^{\tilde{\beta}-\lambda}\right)= \sum_{\tilde{\beta}\in Q_+} \sum_{\vec{m}=(m_1,\ldots,m_n)\in{\mathbb{N}}^n}^{1\leq k\leq N}c(\vec{m})\prod_{i=1}^n (c^+_ic^-_i)^{m_i}\times\\[-1pt] v^{(2\rho,\mu_k)+\sum_{i=1}^n m_i(\nu^+_i-\nu^-_i,\tilde{\beta}-\lambda)-(2\mu_k-\alpha(\vec{m}),\tilde{\beta}-\lambda)-(\mu_k,\alpha(\vec{m}))+\tilde{\tau}^+_{\lambda}(\vec{m})+\tilde{\tau}^-_{\lambda}(\vec{m})}\times\\ \left\langle w_k\middle| E_{\alpha_{i^-_1}}^{m_{i^-_1}}\cdots E_{\alpha_{i^-_n}}^{m_{i^-_n}}\cdot F_{\alpha_{i^+_1}}^{m_{i^+_1}}\cdots F_{\alpha_{i^+_n}}^{m_{i^+_n}}\middle| w_k\right\rangle\cdot J_{\tilde{\beta}}\underline{y}^{\alpha(\vec{m})+\tilde{\beta}-\lambda}. \end{multline} (4.12) Due to the equalities |$\tilde{\tau }^+_{\lambda }(\vec{m})+\tilde{\tau }^-_{\lambda }(\vec{m})= \tau ^+_{\lambda }(\vec{m},\beta )+\tau ^-_{\lambda }(\vec{m},\beta )+\sum _{i=1}^n m_i(\nu ^+_i-\nu ^-_i,\alpha (\vec{m})+\lambda -\beta )$| and |$-(\mu _k,\alpha (\vec{m}))-(2\mu _k-\alpha (\vec{m}),\beta -\lambda -\alpha (\vec{m}))= (\lambda -\beta ,\mu _k)+(\lambda -\beta +\alpha (\vec{m}),\mu _k-\alpha (\vec{m}))$|⁠, the coefficient of |$\underline{y}^{\beta -\lambda }$| in the right-hand side of (4.12) coincides with the left-hand side of (4.11). The equality |$\tilde{\textbf{D}}_V(\tilde{J})=\textrm{tr}_V\left (v^{2(\lambda +\rho )}\right )\cdot \tilde{J}$| follows. 5 Geometric Realization of the Whittaker Vectors in Type A In [3], A. Braverman and M. Finkelberg provided a geometric realization of the universal Verma module over |$U_v(\mathfrak{s}\mathfrak{l}_n)$|⁠, the Shapovalov form on it, and two particular Whittaker vectors |$\mathfrak{k},\mathfrak{w}$| of it via the Laumon based quasiflags’ moduli spaces. The vectors |$\mathfrak{k}$| and |$\mathfrak{w}$| correspond to particular Sevostyanov triples |$(\epsilon ,\mathsf{n},c)$| with the corresponding orientations of the |$A_{n-1}$| Dynkin diagram being equioriented.x` In this section, we generalize their construction by providing a geometric interpretation of all Whittaker vectors and their pairing. 5.1 Laumon spaces First, we recall the set-up of [9]. Let |${\textbf{C}}$| be a smooth projective curve of genus zero. We fix a coordinate |$z$| on |${\textbf{C}}$|⁠, and consider the action of |${\mathbb{C}}^{\times }$| on |${\textbf{C}}$| such that |$v(z)=v^{-2}z$|⁠. We have |${\textbf{C}}^{{\mathbb{C}}^{\times }}=\{0,\infty \}$|⁠. We consider an |$n$|-dimensional vector space |$W$| with a basis |$w_1,\ldots ,w_n$|⁠. This defines a Cartan torus |$T\subset G=\textrm{SL}(n)\subset \textrm{Aut}(W)$|⁠. We also consider its |$2^{n-1}$|-fold cover, the bigger torus |$\widetilde{T}$|⁠, acting on |$W$| as follows: for |$\widetilde{T}\ni \underline{t}=(t_1,\ldots ,t_n)$| we have |$\underline{t}(w_k)=t_k^2w_k$|⁠. Given an |$(n-1)$|-tuple of nonnegative integers |$\underline{d}=(d_1,\ldots ,d_{n-1})\in{\mathbb{N}}^{n-1}$|⁠, we consider the Laumon’s based quasiflags’ space|${\mathfrak Q}_{\underline{d}}$|⁠. It is the moduli space of flags of locally free subsheaves |$0\subset{\mathcal{W}}_1\subset \cdots \subset{\mathcal{W}}_{n-1}\subset{\mathcal{W}}=W\otimes{\mathcal{O}}_{\textbf{C}}$| such that |$\textrm{rk}({\mathcal{W}}_k)=k, \deg ({\mathcal{W}}_k)=-d_k$|⁠, |${\mathcal{W}}_k\subset{\mathcal{W}}$| is a vector subbundle in a neighborhood of |$\infty \in{\textbf{C}}$|⁠, and the fiber of |${\mathcal{W}}_k$| at |$\infty $| equals the span |$\langle w_1,\ldots ,w_k\rangle \subset W$|⁠. It is a smooth connected quasi-projective variety of dimension |$\sum _{i=1}^{n-1} 2d_i$|⁠. The group |$\widetilde{T}\times{\mathbb{C}}^{\times }$| acts naturally on |${\mathfrak Q}_{\underline{d}}$|⁠. The set of fixed points of |$\widetilde{T}\times{\mathbb{C}}^{\times }$| on |${\mathfrak Q}_{\underline{d}}$| is finite and is parametrized by collections |$\widetilde{\underline{d}}$| of nonnegative integers |$(d_{ij})_{1\leq j\leq i\leq n-1}$| such that |$d_i=\sum _{j=1}^i d_{ij}$| and |$d_{kj}\geq d_{ij}$| for |$i\geq k\geq j$|⁠; see [9, 2.11]. Given a collection |$\widetilde{\underline{d}}$| as above, we will denote by |$\widetilde{\underline{d}}+\delta _{ij}$| the collection |$\widetilde{\underline{d}}{}^{\prime}$|⁠, such that |${d{}}^{\prime}_{ij}=d_{ij}+1$|⁠, while |${d{}}^{\prime}_{kl}=d_{kl}$| for |$(k,l)\ne (i,j)$|⁠. By abuse of notation, we use |$\widetilde{\underline{d}}$| to denote the corresponding |$\widetilde{T}\times{\mathbb{C}}^{\times }$|-fixed point in |${\mathfrak Q}_{\underline{d}}$|⁠. For |$i\in \{1,\ldots ,n-1\}$| and |$\underline{d}=(d_1,\ldots ,d_{n-1})$|⁠, we set |$\underline{d}+i:=(d_1,\ldots ,d_i+1,\ldots ,d_{n-1})$|⁠. We have a correspondence |${\mathfrak{E}}_{\underline{d},i}\subset{\mathfrak Q}_{\underline{d}}\times{\mathfrak Q}_{\underline{d}+i}$| formed by the pairs |$({\mathcal{W}}_{\bullet },{\mathcal{W}}^{\prime}_{\bullet })$| such that |${\mathcal{W}}^{\prime}_i\subset{\mathcal{W}}_i$| and |${\mathcal{W}}_j={\mathcal{W}}^{\prime}_j$| for |$j\ne i$|⁠. It is a smooth quasi-projective variety of dimension |$1+\sum _{i=1}^{n-1} 2d_i$|⁠. We denote by |${\textbf{p}}$| (resp. |${\textbf{q}}$|⁠) the natural projection |${\mathfrak{E}}_{\underline{d},i}\to{\mathfrak Q}_{\underline{d}}$| (resp. |${\mathfrak{E}}_{\underline{d},i}\to{\mathfrak Q}_{\underline{d}+i}$|⁠). We also have a map |${\textbf{s}}\colon{\mathfrak{E}}_{\underline{d},i}\to{\textbf{C}}$|⁠, given by |$({\mathcal{W}}_{\bullet },{\mathcal{W}}^{\prime}_{\bullet })\mapsto \operatorname{supp}({\mathcal{W}}_i/{\mathcal{W}}^{\prime}_i)$|⁠. The correspondence |${\mathfrak{E}}_{\underline{d},i}$| comes equipped with a natural line bundle |${\mathcal{L}}_i$| whose fiber at a point |$({\mathcal{W}}_{\bullet },{\mathcal{W}}^{\prime}_{\bullet })$| equals |$\Gamma ({\textbf{C}},{\mathcal{W}}_i/{\mathcal{W}}^{\prime}_i)$|⁠. We denote by |${}^{\prime}M$| the direct sum of equivariant |$K$|-groups: |${}^{\prime}M:=\oplus _{\underline{d}}K^{\widetilde{T}\times{\mathbb{C}}^{\times }}({\mathfrak Q}_{\underline{d}})$|⁠. It is a module over |$K^{\widetilde{T}\times{\mathbb{C}}^{\times }}(\operatorname{pt})={\mathbb{C}}[\widetilde{T}\times{\mathbb{C}}^{\times }]= {\mathbb{C}}[t^{\pm 1}_1,\ldots ,t^{\pm 1}_n,v^{\pm 1}:\ t_1\cdots t_n=1]$|⁠. We define |$M:={}^{\prime}M\otimes _{K^{\widetilde{T}\times{\mathbb{C}}^{\times }}(\operatorname{pt})} \operatorname{Frac}(K^{\widetilde{T}\times{\mathbb{C}}^{\times }}(\operatorname{pt}))\otimes _{{\mathbb{C}}(v)} {\mathbb{C}}(v^{1/\mathsf{N}})$|⁠. It is naturally graded: |$M=\oplus _{\underline{d}}M_{\underline{d}}$|⁠. According to the Thomason localization theorem, restriction to the |$\widetilde{T}\times{\mathbb{C}}^{\times }$|-fixed point set induces an isomorphism of localized |$K$|-groups. The classes of the structure sheaves |$[\widetilde{\underline{d}}]$| of the |$\widetilde{T}\times{\mathbb{C}}^{\times }$|-fixed points |$\widetilde{\underline{d}}$| form a basis in |$\oplus _{\underline{d}}K^{\widetilde{T}\times{\mathbb{C}}^{\times }}({\mathfrak Q}_{\underline{d}}^{\widetilde{T}\times{\mathbb{C}}^{\times }})_{\textrm{loc}}$|⁠. The embedding of a point |$\widetilde{\underline{d}}$| into |${\mathfrak Q}_{\underline{d}}$| is a proper morphism, so the direct image in the equivariant |$K$|-theory is well-defined, and we will denote by |$[\widetilde{\underline{d}}]\in M_{\underline{d}}$| the direct image of the structure sheaf of the point |$\widetilde{\underline{d}}$|⁠. The set |$\{[\widetilde{\underline{d}}]\}$| forms a basis of |$M$|⁠. 5.2 |$U_v(\mathfrak{s}\mathfrak{l}_n)$|-action via Laumon spaces Following Section 4.1, consider the universal Verma module |${\mathcal{V}}$| over |$U_v(\mathfrak{s}\mathfrak{l}_n)$| with |$u_i=v^{\frac{i(i-1)}{2}}t_1\cdots t_i$|⁠, that is, |$L_i(\textbf{1})=v^{\frac{i(i-1)}{2}}t_1\cdots t_i\cdot \textbf{1}$|⁠. We identify |${\mathsf{k}}\simeq \operatorname{Frac}(K^{\widetilde{T}\times{\mathbb{C}}^{\times }}(\operatorname{pt}))\otimes _{{\mathbb{C}}(v)} {\mathbb{C}}(v^{1/\mathsf{N}})$|⁠. Define the following operators on |$M$|⁠: |$E_i:=t_{i+1}^{-1}v^{d_{i+1}-d_i+1-i}{\textbf{p}}_*{\textbf{q}}^*\colon M_{\underline{d}}\to M_{\underline{d}-i},$| |$F_i:=-t_i^{-1}v^{d_i-d_{i-1}+i}{\textbf{q}}_*({\mathcal{L}}_i\otimes{\textbf{p}}^*) \colon M_{\underline{d}}\to M_{\underline{d}+i},$| |$L_i:=t_1\cdots t_iv^{-d_i+\frac{i(i-1)}{2}}\colon M_{\underline{d}}\to M_{\underline{d}},$| |$K_i:=L_{i-1}^{-1}L_i^2L_{i+1}^{-1}=t_{i+1}^{-1}t_iv^{d_{i+1}-2d_i+d_{i-1}-1}\colon M_{\underline{d}}\to M_{\underline{d}}$|⁠. To each |$\widetilde{\underline{d}}$|⁠, we also assign a collection of |$\widetilde{T}\times{\mathbb{C}}^{\times }$|-weights |$s_{ij}:=t_j^2v^{-2d_{ij}}$|⁠. The following result is due to [3] (though our formulas follow [10, 24]). Theorem 5.3. (a) The operators |$\{E_i, F_i, L^{\pm 1}_i\}_{i=1}^{n-1}$| give rise to the action of |$U_v(\mathfrak{s}\mathfrak{l}_n)$| on |$M$|⁠. (b) There is a unique |$U_v(\mathfrak{s}\mathfrak{l}_n)$|-module isomorphism taking |$[{\mathcal{O}}_{{\mathfrak Q}_{\underline{0}}}]$| to |$\textbf{1}$|⁠. (c) The action of |$L_i$| is diagonal in the basis |$\{[\widetilde{\underline{d}}]\}$| and |$L_i([\widetilde{\underline{d}}])=t_1\cdots t_iv^{-d_i+\frac{i(i-1)}{2}}\cdot [\widetilde{\underline{d}}]$|⁠. (d) The matrix coefficients of |$F_i, E_i$| in the fixed point basis |$\{[\widetilde{\underline{d}}]\}$| of |$M$| are as follows: \begin{equation*} F_{i[\widetilde{\underline{d}},\widetilde{\underline{d}}{}^{\prime}]}=-(1-v^2)^{-1}t_i^{-1}v^{d_i-d_{i-1}+i} s_{ij} \prod_{j\ne k\leq i} (1-s_{ij}/s_{ik})^{-1}\prod_{k\leq i-1}(1-s_{ij}/s_{i-1,k}) \end{equation*} if |$\widetilde{\underline{d}^{\prime}}=\widetilde{\underline{d}}+\delta _{ij}$| for certain |$j\leq i$|⁠; \begin{equation*} E_{i[\widetilde{\underline{d}},\widetilde{\underline{d}}{}^{\prime}]}=(1-v^2)^{-1}t_{i+1}^{-1}v^{d_{i+1}-d_i+1-i} \prod_{j\ne k\leq i} (1-s_{ik}/s_{ij})^{-1} \prod_{k\leq i+1}(1-s_{i+1,k}/s_{ij}) \end{equation*} if |$\widetilde{\underline{d}^{\prime}}=\widetilde{\underline{d}}-\delta _{ij}$| for certain |$j\leq i$|⁠. All the other matrix coefficients of |$F_i,E_i$| vanish. 5.4 Geometric realization of the Whittaker vectors Choose a Sevostyanov triple |$(\epsilon ,\mathsf{n},c)$| and let |$e_i:=E_i\prod _{p=1}^{n-1}L_p^{\mathsf{n}_{ip}}=E_iK_{\nu }, \nu :=\sum _{p=1}^{n-1}\mathsf{n}_{ip}\omega _p$|⁠, be the corresponding Sevostyanov generators. Choose |$\underline{a}=(a_1,\ldots ,a_{n-1})\in \{0,1\}^{n-1}$| so that |$a_i=\frac{1+\epsilon _{i-1,i}}{2}=\frac{1-\mathsf{n}_{i-1,i}+\mathsf{n}_{i,i-1}}{2}$| for |$1<i\leq n-1$|⁠, while |$a_1$| equals either |$0$| or |$1$|⁠. Consider the line bundle |${\mathcal{D}}_i$| on |${\mathfrak Q}_{\underline{d}}$| whose fiber at the point |$({\mathcal{W}}_{\bullet })$| equals |$\det R\Gamma ({\textbf{C}},{\mathcal{W}}_i)$|⁠. We also define the line bundle |${\mathcal{D}}^{\underline{a}}$| on |${\mathfrak Q}_{\underline{d}}$| via |${\mathcal{D}}^{\underline{a}}:=\otimes _{i=1}^{n-1}{\mathcal{D}}_i^{-a_i}$|⁠. Note that |${\mathcal{D}}_1$| is a pull-back of the 1st line bundle on the Drinfeld compactification and therefore is trivial, which explains the irrelevance of our choice of |$a_1$|⁠. Finally, we introduce the constants \begin{multline*} X(\underline{d}):=\prod_{i=1}^{n-1} ((1-v^2)c_i)^{d_i} \prod_{p=1}^{n-1}(t_1\cdots t_p)^{-2a_p-\sum_{i=1}^{n-1} d_i \mathsf{n}_{ip}} \prod_{p=1}^{n-1} t_p^{d_{p-1}-2a_pd_p}\times\\ v^{\sum_{i=1}^{n-1} \left((\mathsf{n}_{ii}+i)d_i-2a_{i+1}d_id_{i+1}+\frac{d_i(d_i-1)}{2}(\mathsf{n}_{ii}+2a_i+1)\right)+ \sum_{i<j} \mathsf{n}_{ji}d_id_j-\sum_{i,p=1}^{n-1} \frac{p(p-1)}{2} d_i\mathsf{n}_{ip}}. \end{multline*} The following is the key result of this section. Theorem 5.5. Define |$\theta _{\underline{d}}:=X(\underline{d})\cdot [{\mathcal{D}}^{\underline{a}}]\in M_{\underline{d}}$| and set |$\theta :=\sum _{\underline{d}}\theta _{\underline{d}}$|⁠. Then |$e_i(\theta )=c_i\cdot \theta $| for any |$1\leq i\leq n-1$|⁠. Remark 5.6. (a) Due to Theorem 5.3, this provides a geometric realization of all Whittaker vectors (associated with Sevostyanov triples) of the universal Verma module |${\mathcal{V}}$| over |$U_v(\mathfrak{s}\mathfrak{l}_n)$|⁠. (b) It is straightforward to verify that |$\theta $| does not depend on the choice of |$a_1\in \{0,1\}$|⁠. Proof. According to the Bott–Lefschetz formula, we have the following: |$\theta =\sum _{\widetilde{\underline{d}}} a_{\widetilde{\underline{d}}}\cdot X(\underline{d})\cdot ({\mathcal{D}}^{\underline{a}})_{\mid _{\widetilde{\underline{d}}}}\cdot [\widetilde{\underline{d}}],\ \textrm{where}\ a_{\widetilde{\underline{d}}}=\prod _{w\in T_{\widetilde{\underline{d}}}{\mathfrak Q}_{\underline{d}}}(1-w)^{-1};$| |$\frac{a_{\widetilde{\underline{d}{}^{\prime}}}}{a_{\widetilde{\underline{d}}}}({\textbf{p}}_*{\textbf{q}}^*)_{[\widetilde{\underline{d}{}^{\prime}},\widetilde{\underline{d}}]}= ({\textbf{q}}_*{\textbf{p}}^*)_{[\widetilde{\underline{d}},\widetilde{\underline{d}{}^{\prime}}]}.$| According to Theorem 5.3(c, d), we have |$({\textbf{q}}_*{\textbf{p}}^*)_{[\widetilde{\underline{d}},\widetilde{\underline{d}}+\delta _{ij}]}= \frac{1}{1-v^2}\prod _{j\ne k\leq i} (1-s_{ij}/s_{ik})^{-1}\prod _{k\leq i-1}(1-s_{ij}/s_{i-1,k});$| |$(\prod _{p=1}^{n-1}L_p^{\mathsf{n}_{ip}})[\widetilde{\underline{d}}+\delta _{ij}]= v^{-\mathsf{n}_{ii}}\prod _{p=1}^{n-1}\left (t_1\cdots t_pv^{-d_p+\frac{p(p-1)}{2}}\right )^{\mathsf{n}_{ip}}\cdot [\widetilde{\underline{d}}+\delta _{ij}],$| where |$s_{ij}=t_j^2v^{-2d_{ij}}$| as before. Finally, we also have |$({\mathcal{D}}^{\underline{a}})_{\mid _{\widetilde{\underline{d}}+\delta _{ij}}}/({\mathcal{D}}^{\underline{a}})_{\mid _{\widetilde{\underline{d}}}}=s_{ij}^{a_i}$|⁠. Therefore, it suffices to prove the following equality for any |$\widetilde{\underline{d}}$| and |$1\leq i\leq n-1$|⁠: \begin{equation*} \frac{X(\underline{d}+i)}{X(\underline{d})} \frac{t_{i+1}^{-1}v^{d_{i+1}-d_i-i}}{1-v^2} v^{-\mathsf{n}_{ii}} \prod_{p=1}^{n-1}\left(t_1\cdots t_pv^{-d_p+\frac{p(p-1)}{2}}\right)^{\mathsf{n}_{ip}}\cdot \sum_{j\leq i}s_{ij}^{a_i} \frac{\prod_{k\leq i-1}(1-s_{ij}/s_{i-1,k})}{\prod_{k\leq i}^{k\ne j}(1-s_{ij}/s_{ik})}=c_i. \end{equation*} Lemma 5.7. For any |$\widetilde{\underline{d}}$| and |$1\leq i\leq n-1$|⁠, the following equality holds: \begin{equation} \sum_{j\leq i}s_{ij}^{a_i} \frac{\prod_{k\leq i-1}(1-s_{ij}/s_{i-1,k})} {\prod_{k\leq i}^{k\ne j}(1-s_{ij}/s_{ik})}=(t_i^2v^{2d_{i-1}-2d_i})^{a_i}. \end{equation} (5.1) Proof. First, let us rewrite the left-hand side of (5.1) as \begin{equation*} \frac{s_{i1}\cdots s_{ii}}{s_{i-1,1}\cdots s_{i-1,i-1}}\cdot \sum_{j\leq i} s_{ij}^{a_i-1}\frac{\prod_{k=1}^{i-1} (s_{i-1,k}-s_{ij})}{\prod_{k\leq i}^{k\ne j} (s_{ik}-s_{ij})}. \end{equation*} If |$a_i=1$|⁠, then the above sum |$\sum _{j\leq i} s_{ij}^{a_i-1}\frac{\prod _{k=1}^{i-1} (s_{i-1,k}-s_{ij})}{\prod _{k\leq i}^{k\ne j} (s_{ik}-s_{ij})}$| is a rational function in |$\{s_{ij}\}_{j=1}^i$| of degree |$0$| and without poles, hence, a constant. To evaluate this constant, let |$s_{ii}\to \infty $|⁠, in which case the 1st |$i-1$| summands tend to zero, while the last one tends to |$1$|⁠. Hence, this constant is |$1$|⁠, and the left-hand side of (5.1) equals |$\frac{s_{i1}\cdots s_{ii}}{s_{i-1,1}\cdots s_{i-1,i-1}}=t_i^2v^{2d_{i-1}-2d_i}$|⁠. If |$a_i=0$|⁠, then |$\sum _{j\leq i} s_{ij}^{-1}\frac{\prod _{k=1}^{i-1} (s_{i-1,k}-s_{ij})}{\prod _{k\leq i}^{k\ne j} (s_{ik}-s_{ij})}- \sum _{j\leq i} s_{ij}^{-1}\frac{\prod _{k=1}^{i-1} s_{i-1,k}}{\prod _{k\leq i}^{k\ne j} (s_{ik}-s_{ij})}=0$| as the left-hand side is a rational function in |$\{s_{ij}\}_{j=1}^i$| of degree -|$1$| and without poles. Thus, the left-hand side of (5.1) equals |$\sum _{j\leq i}\frac{\prod _{k\leq i}^{k\ne j} s_{ik}}{\prod _{k\leq i}^{k\ne j} (s_{ik}-s_{ij})}$|⁠. This is a rational function in |$\{s_{ij}\}_{j=1}^i$| of degree |$0$| and without poles, hence, a constant. Specializing |$s_{ii}\mapsto 0$|⁠, we see that this constant equals |$1$| (as the 1st |$i-1$| summands specialize to |$0$|⁠, while the last one specializes to |$1$|⁠). Due to Lemma 5.7, it remains to verify \begin{equation*} \frac{X(\underline{d}+i)}{X(\underline{d})}=(1-v^2)c_i t_{i+1} v^{d_i-d_{i+1}+i} v^{\mathsf{n}_{ii}} \prod_{p=1}^{n-1}\left(t_1\cdots t_pv^{-d_p+\frac{p(p-1)}{2}}\right)^{-\mathsf{n}_{ip}}(t_i^2v^{2d_{i-1}-2d_i})^{-a_i}, \end{equation*} which is straightforward. This completes our proof of Theorem 5.5. Remark 5.8. Note that if |$\epsilon _{i,i+1}=-1$| (resp. |$\epsilon _{i,i+1}=1$|⁠) for all |$i$|⁠, then |$\theta $| is a linear combination of |$[{\mathcal{O}}_{{\mathfrak Q}_{\underline{d}}}]$| (resp. |$[{\mathcal{D}}_{\underline{d}}^{-1}]$| with |${\mathcal{D}}_{\underline{d}}:=\otimes _{i=1}^{n-1}{\mathcal{D}}_i$|⁠). These are exactly the two cases considered in [3]. 5.9 Geometric realization of the |$J$|-function Recall the Shapovalov form |$(\cdot ,\cdot )$| on the universal Verma module |${\mathcal{V}}$|⁠, which is a unique nondegenerate symmetric bilinear form on |${\mathcal{V}}$| with values in |${\mathsf{k}}\simeq \operatorname{Frac}(K^{\widetilde{T}\times{\mathbb{C}}^{\times }}(\operatorname{pt}))\otimes _{{\mathbb{C}}(v)} {\mathbb{C}}(v^{1/\mathsf{N}})$| characterized by |$(\textbf{1},\textbf{1})=1$| and |$(xw,w^{\prime})=(w,\widetilde{\sigma }(x)w^{\prime})$| for all |$w,w^{\prime}\in{\mathcal{V}}, x\in U_v(\mathfrak{s}\mathfrak{l}_n)$|⁠, where |$\widetilde{\sigma }$| is the antiautomorphism of |$U_v(\mathfrak{s}\mathfrak{l}_n)$| determined by |$\widetilde{\sigma }(E_i)=F_i, \widetilde{\sigma }(F_i)=E_i, \widetilde{\sigma }(L_i)=L_i$|⁠. Identifying |${\mathcal{V}}\cong M$| via Theorem 5.3(b), a geometric expression for the Shapovalov form was obtained in [3, Proposition 2.29] (note that our formula differs from the one of [3] as we use a slightly different action of |$U_v(\mathfrak{s}\mathfrak{l}_n)$|⁠). Proposition 5.10. If |$\underline{d}\ne \underline{d}^{\prime}$|⁠, then |$M_{\underline{d}}$| is orthogonal to |$M_{\underline{d}^{\prime}}$|⁠. For |${\mathcal{F}},{\mathcal{F}}^{\prime}\in M_{\underline{d}}$|⁠, we have \begin{equation} ({\mathcal{F}},{\mathcal{F}}^{\prime})=(-1)^{\sum_{i=1}^{n-1}d_i}v^{\sum_{i=1}^{n-1} (d_id_{i+1}-d_i^2+(1-2i)d_i)} \prod_{i=1}^n t_i^{d_i-d_{i-1}}\cdot [R\Gamma({\mathfrak Q}_{\underline{d}}, {\mathcal{F}}\otimes{\mathcal{F}}^{\prime}\otimes{\mathcal{D}}_{\underline{d}})], \end{equation} (5.2) where |${\mathcal{D}}_{\underline{d}}=\otimes _{i=1}^{n-1}{\mathcal{D}}_i$| as in Remark 5.8. Given a pair of Sevostyanov triples |$(\epsilon ^{\pm }, \mathsf{n}^{\pm }, c^{\pm })$|⁠, choose the corresponding |$\underline{a}^{\pm }\in \{0,1\}^{n-1}$| and |$X(\underline{d})^{\pm }\in \operatorname{Frac}(K^{\widetilde{T}\times{\mathbb{C}}^{\times }}(\operatorname{pt}))$|⁠, and define vectors |$\theta ^{\pm }_{\underline{d}}:=X(\underline{d})^{\pm }[{\mathcal{D}}^{\underline{a}^{\pm }}]\in M_{\underline{d}}$| as in Section 5.4. Consider the following generating function: \begin{equation} {\mathfrak{J}}={\mathfrak{J}}(y_1,\ldots,y_{n-1}):=\prod_{i=1}^{n-1}y_i^{-\frac{\log(t_1\cdots t_i)}{\log(v)}-\frac{i(n-1)}{2}}\cdot \sum_{\underline{d}}\left(\theta^+_{\underline{d}},\theta^-_{\underline{d}}\right)y_1^{d_1}\cdots y_{n-1}^{d_{n-1}}. \end{equation} (5.3) Due to (5.2), the coefficient |$(\theta ^+_{\underline{d}},\theta ^-_{\underline{d}})$| equals \begin{multline*} (-1)^{\sum_{i=1}^{n-1}d_i}v^{\sum_{i=1}^{n-1} (d_id_{i+1}-d_i^2+(1-2i)d_i)} X(\underline{d})^+X(\underline{d})^-\prod_{i=1}^n t_i^{d_i-d_{i-1}}\cdot [R\Gamma({\mathfrak Q}_{\underline{d}}, \otimes_{i=1}^{n-1}{\mathcal{D}}_i^{1-a^+_i-a^-_i})]. \end{multline*} The following result is an immediate consequence of Theorems 4.9, 5.3, and Remark 5.6. Theorem 5.11. |${\mathfrak{J}}$| is an eigenfunction of the type |$A_{n-1}$| modified quantum difference Toda system |${\mathcal{T}}(\pm \epsilon ^{\pm },\pm \mathsf{n}^{\pm },c^{\pm })$|⁠. In particular, for |$\textbf{D}_1$| computed explicitly in (3.5), we have \begin{equation} \textbf{D}_1({\mathfrak{J}})=\left(v^{n-1}\sum_{i=1}^n t_i^2\right)\cdot{\mathfrak{J}}. \end{equation} (5.4) Remark 5.12. (a) There is an algebra isomorphism determined by |$E_i\mapsto \bar{E}_i, F_i\mapsto \bar{F}_i, L_i\mapsto \bar{L}^{-1}_i$|⁠. Note that |$\sigma =\varsigma \circ \widetilde{\sigma }$| (with |$\sigma $| defined in Section 4.1) and the action of |$U_v(\mathfrak{s}\mathfrak{l}_n)$| on |$\bar{{\mathcal{V}}}$| (as a |$\varsigma $|-pull-back of |$U_{v^{-1}}(\mathfrak{s}\mathfrak{l}_n)$|-action) is isomorphic to |${\mathcal{V}}$|⁠. This implies that the Shapovalov form on |${\mathcal{V}}$| is identified with the |${\mathsf{k}}$|-bilinear form on |${\mathcal{V}}\times \bar{{\mathcal{V}}}$| of Section 4.1. (b) Under the identification of part (a), the Whittaker vector of |$\bar{{\mathcal{V}}}$| associated with a Sevostyanov triple |$(\epsilon ^-,\mathsf{n}^-,c^-)$| becomes the Whittaker vector of |${\mathcal{V}}$| associated with the Sevostyanov triple |$(-\epsilon ^-,-\mathsf{n}^-,c^-)$|⁠. This explains the appearance of the sign ‘–’ in front of |$\epsilon ^-,\mathsf{n}^-$| in Theorem 5.11. 5.13 B. Feigin’s viewpoint via |$U_v(L\mathfrak{s}\mathfrak{l}_n)$|-action According to [24, Theorem 2.12] (see also [10, Theorem 12.7]), the action of |$U_v(\mathfrak{s}\mathfrak{l}_n)$| on |$M$| can be extended to an action of the quantum loop algebra |$U_v(L\mathfrak{s}\mathfrak{l}_n)$| on |$M$| (actually, this action factors through the one of |$U_v(\mathfrak{g}\mathfrak{l}_n)$|⁠, extending the |$U_v(\mathfrak{s}\mathfrak{l}_n)$|-action from Theorem 5.3, via the evaluation homomorphism |$\textrm{ev}\colon U_v(L\mathfrak{s}\mathfrak{l}_n)\to U_v(\mathfrak{g}\mathfrak{l}_n)$|⁠). In particular, loop generators |$\{e_{i,r},f_{i,r}\}_{1\leq i\leq n-1}^{r\in{\mathbb{Z}}}$| (see [24, 2.10]) act via \begin{equation*} e_{i,r}=t_{i+1}^{-1}v^{d_{i+1}-d_i+1-i}{\textbf{p}}_*((v^i{\mathcal{L}}_i)^{\otimes r}\otimes{\textbf{q}}^*)\colon M_{\underline{d}}\to M_{\underline{d}-i}, \end{equation*} \begin{equation*} f_{i,r}=-t_i^{-1}v^{d_i-d_{i-1}+i}{\textbf{q}}_*({\mathcal{L}}_i\otimes (v^i{\mathcal{L}}_i)^{\otimes r}\otimes{\textbf{p}}^*)\colon M_{\underline{d}}\to M_{\underline{d}+i}. \end{equation*} Note that |$e_{i,0}=E_i$| and |$f_{i,0}=F_i$|⁠. Following [3], define |${\mathfrak{k}}\in M^{\wedge }:=\prod _{\underline{d}} M_{\underline{d}}\cong{\mathcal{V}}^{\wedge }$| via \begin{equation} {\mathfrak{k}}:=\sum_{\underline{d}} {\mathfrak{k}}_{\underline{d}}\ \textrm{with}\ {\mathfrak{k}}_{\underline{d}}:=[{\mathcal{O}}_{{\mathfrak Q}_{\underline{d}}}]\in M_{\underline{d}}. \end{equation} (5.5) Proposition 5.14. (a) For any |$1\leq i\leq n-1$|⁠, we have |$e_{i,0}L_i^{-1}L_{i+1}({\mathfrak{k}})=\frac{v}{1-v^2}{\mathfrak{k}}$|⁠. (b) For any |$1\leq i\leq n-1$|⁠, we have |$e_{i,1}L_{i-1}^2L_i^{-3}L_{i+1}({\mathfrak{k}})=\frac{v^{5-i}}{1-v^2}{\mathfrak{k}}$|⁠. Proof. Part (a) follows from Theorem 5.5 (see also [10, Proposition 12.21]). The proof of part (b) is completely analogous to our proof of Theorem 5.5 (see also [10, Remark 12.22(b)]). Let us now explain the relation between Proposition 5.14 regarding the “eigen-property” of the (geometrically) simplest Whittaker vector |${\mathfrak{k}}$| and the geometric description of the general Whittaker vectors from Theorem 5.5. In what follows, we will view the line bundle |${\mathcal{D}}_i^{\pm 1}$| as an endomorphism of |$M$| given by the multiplication by |$[{\mathcal{D}}_i^{\pm 1}]$|⁠. Proposition 5.15. We have the following equalities in |$\textrm{End}(M)$|⁠: (a) |${\mathcal{D}}_i e_{j,0} {\mathcal{D}}_i^{-1}=e_{j,0}$| for |$j\ne i$|⁠, (b) |${\mathcal{D}}_i e_{i,0} {\mathcal{D}}_i^{-1}=v^{-i} e_{i,1}$|⁠. Proof. According to [8, Corollary 6.5(a)], the operator |${\mathcal{D}}_i$| is diagonal in the fixed point basis |$\{[\widetilde{\underline{d}}]\}$|⁠, and the eigenvalue at |$[\widetilde{\underline{d}}]$| is equal to |$\prod _{k=1}^i t_k^{2(1-d_{ik})}v^{d_{ik}(d_{ik}-1)}$|⁠. Part (a) follows as |$e_{j,0}\colon M_{\underline{d}}\to M_{\underline{d}-j}$|⁠. Likewise, the only nonzero matrix coefficients of |${\mathcal{D}}_i e_{i,0} {\mathcal{D}}_i^{-1}$| are given by |${{\mathcal{D}}_i e_{i,0} {\mathcal{D}}_i^{-1}}_{[\widetilde{\underline{d}},\widetilde{\underline{d}}-\delta _{ij}]}= t_j^2v^{-2(d_{ij}-1)}\cdot e_{i,0 [\widetilde{\underline{d}},\widetilde{\underline{d}}-\delta _{ij}]}= v^2s_{ij}\cdot e_{i,0 [\widetilde{\underline{d}},\widetilde{\underline{d}}-\delta _{ij}]}$|⁠, where |$s_{ij}=t_j^2v^{-2d_{ij}}$| as before. According to [24, Proposition 2.15], the only non-zero matrix coefficients of |$e_{i,1}$| in the fixed point basis are given by |$e_{i,1 [\widetilde{\underline{d}},\widetilde{\underline{d}}-\delta _{ij}]}= v^{i+2}s_{ij}\cdot e_{i,0 [\widetilde{\underline{d}},\widetilde{\underline{d}}-\delta _{ij}]}$|⁠. Part (b) follows. Corollary 5.16. For any |$\underline{a}=(a_1,\ldots ,a_{n-1})\in \{0,1\}^{n-1}$|⁠, the following holds in |$\textrm{End}(M)$|⁠: \begin{equation} ({\mathcal{D}}^{\underline{a}})^{-1} e_{i,0} {\mathcal{D}}^{\underline{a}}= \begin{cases} e_{i,0}, & \mbox{if}\,\, a_i=0,\\ v^{-i}e_{i,1}, & \mbox{if } a_i=1. \end{cases} \end{equation} (5.6) For |$\underline{a}\in \{0,1\}^{n-1}$|⁠, define |${\mathfrak{k}}^{\underline{a}}\in M^{\wedge }$| via \begin{equation} {\mathfrak{k}}^{\underline{a}}:=\sum_{\underline{d}} {\mathfrak{k}}^{\underline{a}}_{\underline{d}}\ \,\, \textrm{with}\,\,\, {\mathfrak{k}}^{\underline{a}}_{\underline{d}}:=\prod_{i=1}^{n-1}(t_1\cdots t_i)^{-2a_i} \cdot [{\mathcal{D}}^{\underline{a}}]\in M_{\underline{d}}. \end{equation} (5.7) Note that |${\mathfrak{k}}^{\underline{0}}={\mathfrak{k}}$|⁠. The special case of Theorem 5.5 follows immediately from Proposition 5.14. Proposition 5.17. |${\mathfrak{k}}^{\underline{a}}\in M^{\wedge }\cong{\mathcal{V}}^{\wedge }$| is the Whittaker vector corresponding to the Sevostyanov triple |$(\epsilon ,\mathsf{n},c)$| with |$\epsilon _{i,i+1}=2a_{i+1}-1, \mathsf{n}_{ij}=\delta _{j,i+1}-(1+2a_i)\delta _{j,i}+2a_i\delta _{j,i-1},\ c_i=\frac{v^{1+a_i(4-2i)}}{1-v^2}.$| Proof. Since |${\mathfrak{k}}^{\underline{a}}_{\underline{0}}=[{\mathcal{O}}_{{\mathfrak Q}_{\underline{0}}}]$|⁠, it remains to verify the following equality for any |$1\leq i\leq n-1$|⁠: \begin{equation} e_{i,0}L_{i-1}^{2a_i}L_i^{-1-2a_i}L_{i+1}([{\mathcal{D}}^{\underline{a}}])=\frac{v^{1+a_i(4-2i)}}{1-v^2}\cdot [{\mathcal{D}}^{\underline{a}}]. \end{equation} (5.8) This follows by combining formula (5.6) with Proposition 5.14. Remark 5.18. We note that the above operator of multiplication by |$[{\mathcal{D}}_k]$| can be interpreted entirely algebraically as a product of the Drinfeld Casimir element of the subalgebra |$U_v(\mathfrak{s}\mathfrak{l}_k)\subset U_v(\mathfrak{s}\mathfrak{l}_n)$| and a certain Cartan element, due to [8, Corollary 6.5(b)]. In loc.cit., the authors choose to work with the action of |$U_v(\mathfrak{g}\mathfrak{l}_n)$| instead of |$U_v(\mathfrak{s}\mathfrak{l}_n)$|⁠, which results in shorter formulas. We conclude this section by generalizing the construction of [4, Theorem 6.12]. In loc.cit., the authors established an edge-weight path model for the type |$A_{n-1}$| Whittaker vector associated with a particular Sevostyanov triple |$(\epsilon ,\mathsf{n},c)$| with |$\epsilon _{i,i+1}=1\ (1\leq i\leq n-2)$| and |$\mathsf{n}_{ij}=(i-1)(\delta _{j,i+1}-2\delta _{j,i}+\delta _{j,i-1})$|⁠. More generally, their construction can be applied to Whittaker vectors associated with |$(\epsilon ,\mathsf{n},c)$| satisfying |$\epsilon _{1,2}=\epsilon _{2,3}=\ldots =\epsilon _{n-2,n-1}$| (corresponding to an equioriented |$A_{n-1}$| Dynkin diagram). In particular, identifying |${\mathcal{V}}\cong M$|⁠, we obtain the following edge-weight path model for the Whittaker vector |${\mathfrak{k}}\in M^{\wedge }$| of (5.5). Proposition 5.19. The following equality holds: \begin{equation} {\mathfrak{k}}=\sum_{\beta\in Q_+} \left(\frac{v}{1-v^2}\right)^{|\beta|}\sum_{{\textbf{P}}\in{\mathcal{P}}_{\beta}} y({\textbf{P}})\cdot |{\textbf{P}}\rangle, \end{equation} (5.9) where we use the following notations: |$|\beta |:=\sum _{i=1}^{n-1} m_i$| for |$\beta =\sum _{i=1}^{n-1} m_i\alpha _i\in Q_+$|⁠, the set |${\mathcal{P}}_{\beta }$| consists of all paths |${\textbf{P}}=(p_0,\ldots ,p_N)$| such that |$p_0=0,\ p_N=\beta ,$| and |$p_k-p_{k-1}=\alpha _{i_k} (1\leq i_k\leq n-1)$| for all |$1\leq k\leq N$|⁠, for |${\textbf{P}}=(p_0,\ldots ,p_N)\in{\mathcal{P}}_{\beta }$|⁠, the vector |$|{\textbf{P}}\rangle \in M$| is defined as |$|{\textbf{P}}\rangle :=f_{i_N}f_{i_{N-1}}\cdots f_{i_1}([{\mathcal{O}}_{{\mathfrak Q}_{\underline{0}}}])$| with |$f_i:=L_iL_{i+1}^{-1}F_i$|⁠, for |${\textbf{P}}=(p_0,\ldots ,p_N)\in{\mathcal{P}}_{\beta }$| with |$p_k-p_{k-1}=\alpha _{i_k}$|⁠, the coefficient |$y({\textbf{P}})$| edge-factorizes as |$y({\textbf{P}})=\prod _{k=1}^N \frac{1}{{\mathfrak{v}}^{(i_k)}(p_k)}$|⁠, where |${\mathfrak{v}}^{(i)}(\gamma )=v^{\tau _i(\gamma )} {\mathfrak{v}}(\gamma )$| with |$\tau _i(\gamma )=(\lambda +\rho -\gamma ,\omega _{i-1}-\omega _{i+1})$| and |${\mathfrak{v}}(\gamma )=(v-v^{-1})^{-2} \sum _{i=0}^{n-1}\left (v^{2(\lambda +\rho ,\omega _{i+1}-\omega _i)}-v^{2(\lambda +\rho -\gamma ,\omega _{i+1}-\omega _i)}\right )$| for |$1\leq i\leq n-1$| and |$\gamma \in Q_+$|⁠. Here |$\omega _i$| is the |$i$|-th fundamental weight of |$\mathfrak{s}\mathfrak{l}_n$| as before, and we set |$\omega _0:=0,\ \omega _n:=0$|⁠. Noteworthy, there seems to be no such straightforward edge-weight path model for a general type |$A$| Whittaker vector. Nevertheless, one can fix this by changing the above definition of |$|{\textbf{P}}\rangle $| with the help of the quantum loop algebra |$U_v(L\mathfrak{s}\mathfrak{l}_n)$| in spirit of Proposition 5.15 and Corollary 5.16. This is based on the following result. Proposition 5.20. For any |$\underline{a}=(a_1,\ldots ,a_{n-1})\in \{0,1\}^{n-1}$|⁠, the following holds in |$\textrm{End}(M)$|⁠: \begin{equation} {\mathcal{D}}^{\underline{a}} f_{i,0} ({\mathcal{D}}^{\underline{a}})^{-1}= \begin{cases} f_{i,0}, & \mbox{if}\ a_i=0, \\ v^{-i}f_{i,1}, & \mbox{if } a_i=1, \end{cases} \end{equation} (5.10) where |${\mathcal{D}}^{\underline{a}}$| denotes an endomorphism of |$M$| given by the multiplication by |$[{\mathcal{D}}^{\underline{a}}]$|⁠. Proof. The proof is completely analogous to that of Proposition 5.15. Recall the element |${\mathfrak{k}}^{\underline{a}}\in M^{\wedge }$| of (5.7). Since |$[{\mathcal{D}}^{\underline{a}}]={\mathcal{D}}^{\underline{a}}({\mathfrak{k}})$|⁠, we obtain the following edge-weight path model for |${\mathfrak{k}}^{\underline{a}}$|⁠. Proposition 5.21. The following equality holds: \begin{equation} {\mathfrak{k}}^{\underline{a}}=\prod_{i=1}^{n-1}(t_1\cdots t_i)^{-2a_i} \cdot \sum_{\beta\in Q_+} \left(\frac{v}{1-v^2}\right)^{|\beta|}\sum_{{\textbf{P}}\in{\mathcal{P}}_{\beta}} y({\textbf{P}})\cdot |{\textbf{P}}\rangle^{\underline{a}}, \end{equation} (5.11) where for a path |${\textbf{P}}=(p_0,\ldots ,p_N)\in{\mathcal{P}}_{\beta }$| with |$p_k-p_{k-1}=\alpha _{i_k}$| we set \begin{equation*} |{\textbf{P}}\rangle^{\underline{a}}:=f^{\underline{a}}_{i_N}f^{\underline{a}}_{i_{N-1}}\cdots f^{\underline{a}}_{i_1}([{\mathcal{O}}_{{\mathfrak Q}_{\underline{0}}}]) \ \textrm{with}\ f^{\underline{a}}_i:= \begin{cases} L_iL_{i+1}^{-1}f_{i,0}, & \mbox{if}\ a_i=0, \\ v^{-i}L_iL_{i+1}^{-1}f_{i,1}, & \mbox{if}\ a_i=1. \end{cases} \end{equation*} Proof. Follows by combining Propositions 5.19 and 5.20. According to Proposition 5.17, |${\mathfrak{k}}^{\underline{a}}$| is the Whittaker vector corresponding to the Sevostyanov triple |$(\epsilon ,\mathsf{n},c)$| with |$\epsilon _{i,i+1}=2a_{i+1}-1, \mathsf{n}_{ij}=\delta _{j,i+1}-(1+2a_i)\delta _{j,i}+2a_i\delta _{j,i-1}, c_i=\frac{v^{1+a_i(4-2i)}}{1-v^2}$|⁠. As |$\underline{a}\in \{0,1\}^{n-1}$| varies, we get all possible orientations |$\textrm{Or}$| of |$\textrm{Dyn}(\mathfrak{s}\mathfrak{l}_n)=A_{n-1}$| (here |$\textrm{Or}$| is determined by |$\epsilon $|⁠). Since it is clear how the edge-weight path model gets modified once we change |$\mathsf{n},c$| (while |$\epsilon $| is kept fixed), cf. [7, (3.8, 3.9)], Proposition 5.21 provides an edge-weight path model for a general type |$A$| Whittaker vector. A Proof of Proposition 3.11 Given two pairs of type |$A_{n-1}$| Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| and |$(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| such that |$\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}=\tilde{\epsilon }^+_{i,i+1}-\tilde{\epsilon }^-_{i,i+1}$| for |$1\leq i\leq n-2$|⁠, we will prove that there exist constants |$\{r_{ij},r_i\}_{1\leq i\leq j\leq n}$| such that the function \begin{equation} F=F({\mathsf{w}}_1,\ldots,{\mathsf{w}}_n):= \exp\left(\sum_{1\leq i\leq j\leq n} r_{ij}\log({\mathsf{w}}_i)\log({\mathsf{w}}_j)+\sum_{1\leq i\leq n} r_i\log({\mathsf{w}}_i)\right) \end{equation} (A1) satisfies the equality \begin{equation} F^{-1}\mathsf{H}(\epsilon^{\pm},\mathsf{n}^{\pm},c^{\pm})F=\mathsf{H}(\tilde{\epsilon}^{\pm},\tilde{\mathsf{n}}^{\pm},\tilde{c}^{\pm}). \end{equation} (A2) We will view this as an equality in |${\mathcal{A}}_n$| (rather than |$\bar{{\mathcal{A}}}_n$|⁠), treating |$\mathsf{H}$| of (3.8) as elements of |${\mathcal{A}}_n$|⁠. This will immediately imply the result of Proposition 3.11. Set |$\hbar :=\log (v)$|⁠. |$\bullet $| First, we note that the terms without |${\mathsf{D}}_i$|’s are the same (and equal to |$\sum _{j=1}^n {\mathsf{w}}_j^{-2}$|⁠) both in |$F^{-1}\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })F$| and |$\mathsf{H}(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠, independently of our choice of constants |$\{r_{ij},r_i\}$|⁠. |$\bullet $| Second, we will match the terms with |$\{\frac{{\mathsf{D}}_i}{{\mathsf{D}}_{i+1}}\}_{i=1}^{n-1}$| appearing in |$F^{-1}\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })F$| and |$\mathsf{H}(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. Their equality is equivalent to the following system of equations on |$\{r_{ij}\}$|⁠: \begin{equation} \frac{\mathsf{m}_{ij}-\tilde{\mathsf{m}}_{ij}}{\hbar}= \begin{cases} r_{ji}-r_{j,i+1}, &\ \textrm{if}\ 1\leq j< i\\ r_{ij}-r_{i+1,j}, &\ \textrm{if}\ i+2\leq j\leq n\\ 2r_{ii}-r_{i,i+1}, &\ \textrm{if}\ j=i\\ r_{i,i+1}-2r_{i+1,i+1}, &\ \textrm{if}\ j=i+1 \end{cases} \end{equation} (A3) and the following system of equations on |$\{r_i\}$|⁠: \begin{equation} r_i-r_{i+1}=\hbar(r_{i,i+1}-r_{ii}-r_{i+1,i+1})+\hbar^{-1}\log(\tilde{b}_i/b_i)+ \sum_{k=1}^n(n-k+1/2)(\mathsf{m}_{ik}-\tilde{\mathsf{m}}_{ik}), \end{equation} (A4) where the coefficients |$\mathsf{m}_{ij},\tilde{\mathsf{m}}_{ij}, b_i, \tilde{b}_i$| are defined as in Section 3.9 via |$\mathsf{m}_{ij}:=\sum _{p=j}^{n-1}(\mathsf{n}^-_{ip}-\mathsf{n}^+_{ip}),$| |$\tilde{\mathsf{m}}_{ij}:=\sum _{p=j}^{n-1}(\tilde{\mathsf{n}}^-_{ip}-\tilde{\mathsf{n}}^+_{ip}),$| |$b_i:=(v-v^{-1})^2v^{\mathsf{n}^+_{ii}-\mathsf{n}^-_{ii}}c^+_ic^-_i,$| |$\tilde{b}_i:=(v-v^{-1})^2v^{\tilde{\mathsf{n}}^+_{ii}-\tilde{\mathsf{n}}^-_{ii}}\tilde{c}^+_i\tilde{c}^-_i.$| It suffices to show that (A3) admits a solution, since (A4) obviously admits a solution in terms of |$r_i$| (unique up to a common constant). Pick any |$r_{11}$|⁠. Using the last two cases of (A3), we determine uniquely |$\{r_{i,i+1},r_{i+1,i+1}\}_{i=1}^{n-1}$|⁠. Using the 1st case of (A3), we determine uniquely |$r_{ij}$| for |$j>i+1$|⁠. The resulting collection |$\{r_{ij}\}_{1\leq i\leq j\leq n}$| satisfies the 1st, 3rd, and 4th cases of (A3). It remains to verify that it also satisfies the 2nd case of (A3). We prove this by induction in |$j-i\geq 2$|⁠. (a) If |$j=i+2$|⁠, then |$r_{i,i+2}-r_{i+1,i+2}=(r_{i,i+1}-2r_{i+1,i+1})+(2r_{i+1,i+1}-r_{i+1,i+2})-(r_{i,i+1}-r_{i,i+2})= \hbar ^{-1}(\mathsf{m}_{i,i+1}+\mathsf{m}_{i+1,i+1}-\mathsf{m}_{i+1,i}-\tilde{\mathsf{m}}_{i,i+1}-\tilde{\mathsf{m}}_{i+1,i+1}+\tilde{\mathsf{m}}_{i+1,i})$|⁠. Hence, it remains to prove |$\mathsf{m}_{i,i+1}+\mathsf{m}_{i+1,i+1}-\mathsf{m}_{i+1,i}-\mathsf{m}_{i,i+2}= \tilde{\mathsf{m}}_{i,i+1}+\tilde{\mathsf{m}}_{i+1,i+1}-\tilde{\mathsf{m}}_{i+1,i}-\tilde{\mathsf{m}}_{i,i+2}$|⁠. Since |$\mathsf{m}_{st}-\mathsf{m}_{s,t+1}=\mathsf{n}^-_{st}-\mathsf{n}^+_{st}$|⁠, this is reduced to |$\mathsf{n}^-_{i,i+1}-\mathsf{n}^+_{i,i+1}-\mathsf{n}^-_{i+1,i}+\mathsf{n}^+_{i+1,i}= \tilde{\mathsf{n}}^-_{i,i+1}-\tilde{\mathsf{n}}^+_{i,i+1}-\tilde{\mathsf{n}}^-_{i+1,i}+\tilde{\mathsf{n}}^+_{i+1,i}$|⁠. The latter equality follows from |$\mathsf{n}^{\pm }_{st}-\mathsf{n}^{\pm }_{ts}=\epsilon ^{\pm }_{st}b_{st}$| and our assumption on the triples. (b) If |$j>i+2$|⁠, then |$r_{ij}-r_{i+1,j}=(r_{i,j-1}-r_{i+1,j-1})+(r_{i+1,j-1}-r_{i+1,j})-(r_{i,j-1}-r_{ij})= \hbar ^{-1}(\mathsf{m}_{i,j-1}+\mathsf{m}_{j-1,i+1}-\mathsf{m}_{j-1,i}-\tilde{\mathsf{m}}_{i,j-1}-\tilde{\mathsf{m}}_{j-1,i+1}+\tilde{\mathsf{m}}_{j-1,i})$|⁠. Hence, it remains to prove |$(\mathsf{m}_{i,j-1}-\mathsf{m}_{ij})-(\mathsf{m}_{j-1,i}-\mathsf{m}_{j-1,i+1})= (\tilde{\mathsf{m}}_{i,j-1}-\tilde{\mathsf{m}}_{ij})-(\tilde{\mathsf{m}}_{j-1,i}-\tilde{\mathsf{m}}_{j-1,i+1})$|⁠. Similarly to (a), this is reduced to the proof of |$b_{i,j-1}(\epsilon ^-_{i,j-1}-\epsilon ^+_{i,j-1})= b_{i,j-1}(\tilde{\epsilon }^-_{i,j-1}-\tilde{\epsilon }^+_{i,j-1})$|⁠. The latter follows immediately from the equality |$b_{i,j-1}=0$|⁠. Thus, we have determined a collection of constants |$\{r_{ij},r_i\}_{1\leq i\leq j\leq n}$| satisfying (A3, A4) (this collection is uniquely determined by a choice of |$r_{11},r_1$|⁠; however, we note that the image of |$F$| defined via (A1) in |$\bar{{\mathcal{A}}}_n$| is independent of this choice). |$\bullet $| Finally, it remains to verify that for |$F$| of (A1) with the constants |$r_{ij},r_i$| chosen as above, the terms with |$\frac{{\mathsf{D}}_i}{{\mathsf{D}}_j}\ (j>i+1)$| in |$F^{-1}\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })F$| and |$\mathsf{H}(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| do coincide. First, we note that the conditions |$\epsilon ^{\pm }_{i,i+1}=\ldots =\epsilon ^{\pm }_{j-2,j-1}=\pm 1$| and |$\tilde{\epsilon }^{\pm }_{i,i+1}=\ldots =\tilde{\epsilon }^{\pm }_{j-2,j-1}=\pm 1$| are equivalent under our assumption on the triples. Pick |$j>i+1$| such that either of these equivalent conditions is satisfied. Then the compatibility of the terms with |$\frac{{\mathsf{D}}_i}{{\mathsf{D}}_j}$| is equivalent to the following equality: \begin{multline} \frac{F({\mathsf{w}}_1,\ldots, v{\mathsf{w}}_i,\ldots,v^{-1}{\mathsf{w}}_j,\ldots,{\mathsf{w}}_n)}{F({\mathsf{w}}_1,\ldots,{\mathsf{w}}_n)}=\\ \prod_{k=1}^n {\mathsf{w}}_k^{\sum_{s=i}^{j-1}(\mathsf{m}_{sk}-\tilde{\mathsf{m}}_{sk})}\cdot \prod_{s=i}^{j-1}\frac{\tilde{b}_s}{b_s}\cdot v^{\sum_{i\leq a<b\leq j-1} (\mathsf{n}^-_{ab}-\mathsf{n}^+_{ab}-\tilde{\mathsf{n}}^-_{ab}+\tilde{\mathsf{n}}^+_{ab})+\sum_{k=1}^n \sum_{s=i}^{j-1} \frac{n+1-2k}{2}(\mathsf{m}_{sk}-\tilde{\mathsf{m}}_{sk})}. \end{multline} (A5) We prove this by induction in |$j-i$|⁠. Note that the |$j=i+1$| counterpart of (A5) is just the compatibility of the terms with |$\frac{{\mathsf{D}}_i}{{\mathsf{D}}_{i+1}}$|⁠, established in the previous step. Writing the left-hand side of (A5) as a product \begin{equation} \frac{F({\mathsf{w}}_1,\ldots, v{\mathsf{w}}_i,\ldots,v^{-1}\cdot v{\mathsf{w}}_{j-1},v^{-1}{\mathsf{w}}_j,\ldots,{\mathsf{w}}_n)}{F({\mathsf{w}}_1,\ldots,v{\mathsf{w}}_{j-1},v^{-1}{\mathsf{w}}_j,\ldots,{\mathsf{w}}_n)}\cdot \frac{F({\mathsf{w}}_1,\ldots, v{\mathsf{w}}_{j-1},v^{-1}{\mathsf{w}}_j,\ldots,{\mathsf{w}}_n)}{F({\mathsf{w}}_1,\ldots,{\mathsf{w}}_n)} \end{equation} (A6) and applying the induction assumption to both fractions of (A6), it is straightforward to see that we obtain the right-hand side of (A5). Thus, the function |$F$| defined via (A1) with the constants |$\{r_{ij},r_i\}_{1\leq i\leq j\leq n}$| determined in our 2nd step satisfies the equality (A2). This completes our proof of Proposition 3.11. Remark A.1. (a) The proofs of Propositions 3.14, 3.17, and 3.20 are analogous to the above proof of Proposition 3.11. In each case, there exists a unique collection of constants |$\{r_{ij},r_i\}_{1\leq i\leq j\leq n}$| such that the function |$F$| defined via (A1) satisfies the corresponding equality (A2). The way we choose such constants closely follows the above 2nd step in our proof of Proposition 3.11 and is determined by matching up the coefficients of |$\bullet $||$\{{\mathsf{D}}_i/{\mathsf{D}}_{i+1}\}_{i=1}^{n-1}$| and |${\mathsf{D}}_n^2$| for the type |$C_n$|⁠, |$\bullet $||$\{{\mathsf{D}}_i/{\mathsf{D}}_{i+1}\}_{i=1}^{n-1}$| and |${\mathsf{D}}_{n-1}{\mathsf{D}}_n$| for the type |$D_n$|⁠, |$\bullet $||$\{{\mathsf{D}}_i/{\mathsf{D}}_{i+1}\}_{i=1}^{n-1}$| and |${\mathsf{D}}_n$| for the type |$B_n$|⁠. Finally, it remains to check that the function |$F$| defined via (A1) with thus determined |$\{r_{ij},r_i\}_{1\leq i\leq j\leq n}$| conjugates each of the remaining terms appearing in |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| into the one of |$\mathsf{H}(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. This is verified by induction similarly to the above last step in our proof of Proposition 3.11. (b) The proof of Proposition 3.38 is also analogous, but the constants |$r_{ij},r_i$| are determined by matching the terms with |${\mathsf{D}}_1$| and |${\mathsf{D}}_2$|⁠. B Proof of Theorem 3.2 Assume that |$\mathfrak{g}$| is either of the classical type or |$G_2$|⁠. Given two pairs of Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| and |$(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| with |$\vec{\epsilon }=\vec{\tilde{\epsilon }}$| (defined right before Theorem 3.2), we need to show that there exists an automorphism of |${\mathcal{D}}_v(H^{\textrm{ad}})$| that maps |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |${\mathcal{T}}(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. According to our proof of Proposition 3.11 and Remark A.1 (which states that the same argument applies to all classical types and |$G_2$|⁠), there exists a “formal function” |$F$| of the shift operators |$T_{\mu }$| such that conjugation by |$F$| is a well-defined automorphism of |${\mathcal{D}}_v(H^{\textrm{ad}})$| satisfying |$F\textbf{D}_1(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })F^{-1}=\textbf{D}_1(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. It remains to prove the following result. Proposition B.1. For any |$1\leq i\leq n$|⁠, we have |$F\textbf{D}_i(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })F^{-1}=\textbf{D}_i(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. Proof. Recall that |$\textbf{D}_i\in{\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$|⁠, where |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$| is the subalgebra of |${\mathcal{D}}_v(H^{\textrm{ad}})$| generated by |$\{e^{-\alpha _i},T_{\mu }|1\leq i\leq n, \mu \in P\}$|⁠. Let us extend the field |${\mathbb{C}}(v^{1/\mathsf{N}})$| to |${\mathsf{k}}$| and recall the vector space |$N_{\lambda }$| of Section 4.8, which was equipped with a natural |${\mathcal{D}}_v(H^{\textrm{ad}})$|-action. In particular, the subspace |$W_{\lambda }$| of |$N_{\lambda }$| formed by the formal sums |$\left \{\sum _{\beta \in Q_+} a_{\beta } \underline{y}^{\beta -\lambda }|a_{\beta }\in{\mathsf{k}}\right \}$| is |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$|-stable. Moreover, |$X(\underline{y}^{\beta -\lambda })$| contains only |$\underline{y}^{\gamma -\lambda }$| with |$\gamma \geq \beta $| for any |$X\in{\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$|⁠, which we refer to as the “upper-triangular” property of the |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$|-action. In particular, we have \begin{equation} \textbf{D}_i(\epsilon^{\pm},\mathsf{n}^{\pm},c^{\pm})(\underline{y}^{\beta-\lambda}), \textbf{D}_i(\tilde{\epsilon}^{\pm},\tilde{\mathsf{n}}^{\pm},\tilde{c}^{\pm})(\underline{y}^{\beta-\lambda})\in \left(\sum_{k=1}^{N_i}v^{2(\mu^{(i)}_k,\lambda-\beta)}\right)\cdot \underline{y}^{\beta-\lambda} \oplus \bigoplus_{\gamma>\beta}{\mathsf{k}} \underline{y}^{\gamma-\lambda}, \end{equation} (B1) where |$N_i$| is the dimension and |$\{\mu ^{(i)}_k\}_{k=1}^{N_i}$| are the weights (counted with multiplicities) of the |$i$|-th fundamental |$U_v(\mathfrak{g})$|-representation |$V_i$|⁠, while |$v^{(\nu ,\lambda )}\ (\nu \in P)$| is defined as in Section 4.1. Therefore, the action of |$\textbf{D}_1(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| on |$W_{\lambda }$| is upper-triangular with pairwise distinct diagonal matrix coefficients; hence, it is diagonalizable with a simple spectrum. Moreover, the eigenvalues are exactly |$\left \{\sum _{k=1}^{N_1}v^{2(\mu ^{(1)}_k,\lambda -\beta )}|\beta \in Q_+\right \}$|⁠. Remark B.2. Due to Theorem 4.9, the corresponding eigenbasis consists of the |$J$|-functions |$J(\{y_i\}_{i=1}^n)$| associated with |$\{\lambda -\rho -\beta |\beta \in Q_+\}$|⁠, cf. (4.9). Since |$[\textbf{D}_i(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm }),\textbf{D}_1(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })]=0$|⁠, the action of |$\textbf{D}_i(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| on |$W_{\lambda }$| is diagonal in a |$\textbf{D}_1(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|-eigenbasis with the corresponding eigenvalues given by |$\sum _{k=1}^{N_i}v^{2(\mu ^{(i)}_k,\lambda -\beta )}$| (this also follows from Remark B.2 and Theorem 4.9). On the other hand, the action of |$F\textbf{D}_i(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })F^{-1}$| on |$W_{\lambda }$| is also upper-triangular with the same diagonal matrix coefficients and commutes with |$\textbf{D}_1(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| (since |$F\textbf{D}_1(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })F^{-1}=\textbf{D}_1(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| and |$[\textbf{D}_1(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm }), \textbf{D}_i(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })]=0$|⁠). Thus, both |$F\textbf{D}_i(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })F^{-1}$| and |$\textbf{D}_i(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| act diagonally in a |$\textbf{D}_1(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|-eigenbasis and have the same corresponding eigenvalues. The equality |$F\textbf{D}_i(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })F^{-1}=\textbf{D}_i(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| follows. Thus, conjugation by |$F$| maps |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |${\mathcal{T}}(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. Theorem 3.2 follows. C Proof of Theorem 3.3 Following the discussion in Appendix B, consider a basis of |$W_{\lambda }$| in which all |$\textbf{D}_i$| act simultaneously diagonally with the corresponding eigenvalues given by |$\sum _{k=1}^{N_i}v^{2(\mu ^{(i)}_k,\lambda -\beta )}$|⁠. The latter can be viewed as characters |$\chi _i$| of the fundamental representations evaluated at |$v^{2(\lambda -\beta )}$|⁠. Since the point |$v^{2\lambda }\in H({\mathsf{k}})$| is general and the characters |$\{\chi _i\}_{i=1}^n$| are known to be algebraically independent, we immediately obtain part (a) of Theorem 3.3. Part (c) of Theorem 3.3 follows from part (b) as |$\textbf{D}_V(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })\in{\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$| commutes with |$\textbf{D}_1(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| for any finite-dimensional |$U_v(\mathfrak{g})$|-representation |$V$|⁠, due to Lemma 2.8(c). It remains to prove part (b) of Theorem 3.3. The algebra |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$| is |${\mathbb{Z}}$|-graded via |$\deg (T_{\mu })=0$| and |$\deg (e^{-\alpha _i})=-1$|⁠, so that the degree zero component |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})^0$| has a basis |$\{T_{\mu }|\mu \in P\}$|⁠. Note that the degree zero component |$\textbf{D}_i^{(0)}$| of |$\textbf{D}_i$| equals |$\textbf{D}_i^{(0)}=\sum _{k=1}^{N_i} T_{2\mu ^{(i)}_k}$|⁠. Let |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})^0_{ev}$| be the subspace spanned by |$\{T_{2\mu }|\mu \in P\}$|⁠. We also consider a natural action of the Weyl group |$W$| both on |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})^0$| and |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})^0_{ev}$| via |$w(T_{\mu })=T_{w\mu }$| for |$\mu \in P,w\in W$|⁠. Given |$\textbf{D}\in{\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$| that commutes with |$\textbf{D}_1$|⁠, let |$\textbf{D}^{(0)}$| denote its degree zero component. Proposition C.1. We have |$\textbf{D}^{(0)}\in ({\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})^0_{ev})^W$|⁠. The proof of Proposition C.1 is based on the rank |$1$| case, for which we prove a slightly more general result. In type |$A_1$|⁠, the modified quantum difference Toda systems are conjugate to the |$q$|-Toda of [6] with the 1st hamiltonian |$\textbf{D}_1=T_{2\varpi _1}+T_{-2\varpi _1}-(v-v^{-1})^2e^{-\alpha }T_0$|⁠; see (3.6). Lemma C.2. In type |$A_1$|⁠, given |$\textbf{D}\in{\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$| that commutes with |$\textbf{D}^{\prime}=a_r\textbf{D}_1^r+a_{r-1}\textbf{D}_1^{r-1}+\ldots +a_0$| for some |$a_0,\ldots ,a_r\in{\mathbb{Q}}(v^{1/\mathsf{N}})$| with |$a_r\ne 0,r>0$|⁠, |$\textbf{D}$| must be a polynomial in |$\textbf{D}_1$|⁠. Proof. Let |$\textbf{D}=\textbf{D}^{(0)}+e^{-\alpha }\textbf{D}^{(-1)}+\ldots +e^{-s\alpha }\textbf{D}^{(-s)}$| with |$\textbf{D}^{(0)},\ldots ,\textbf{D}^{(-s)}\in{\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})^0$| and |$\textbf{D}^{(-s)}\ne 0$|⁠. We prove the claim by induction in |$s$|⁠. Comparing the degree |$-r-s$| terms in |$\textbf{D}\textbf{D}^{\prime}=\textbf{D}^{\prime}\textbf{D}$|⁠, we immediately get |$\textbf{D}^{(-s)}=c_sT_0$| for some constant |$c_s$|⁠. Replacing |$\textbf{D}$| by |$\textbf{D}-c_s(-(v-v^{-1})^{-2})^s\textbf{D}_1^s$|⁠, we obtain another element of |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$| that commutes with |$\textbf{D}^{\prime}$| and has a smaller value of |$s$|⁠, hence, is a polynomial in |$\textbf{D}_1$| by the induction assumption. Therefore, |$\textbf{D}$| is also a polynomial in |$\textbf{D}_1$|⁠. Proof of Proposition C.1. The result of Proposition C.1 follows immediately from Lemma C.2. Indeed, it suffices to verify the following two claims for any |$1\leq i\leq n$|⁠: (I) the operator |$\textbf{D}^{(0)}$| is invariant with respect to the simple reflection |$s_i$|⁠, (II) every |$\mu $| appearing in |$\textbf{D}^{(0)}$| satisfies |$(\mu , \alpha _i)\in 2{\mathsf{d}}_i{\mathbb{Z}}$|⁠. To prove this, consider a subspace |$W^{\prime}_{\lambda }$| of |$W_{\lambda }$| that consists of |$\left \{\sum _{\beta \in Q_+\backslash{\mathbb{Z}}\alpha _i} a_{\beta } \underline{y}^{\beta -\lambda }|a_{\beta }\in{\mathsf{k}} \right \}$|⁠. It is stable under the action of |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$|⁠, hence, we obtain the action of |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})$| on the quotient |$\bar{W}_{\lambda }:=W_{\lambda }/W^{\prime}_{\lambda }$|⁠. We also specialize |$u_j\mapsto 1$| for |$j\ne i$| (recall that |$u_j$| were used in our definition of |$\lambda $|⁠). As a result, summands with |$e^{-\alpha _j}\ (j\ne i)$| in |$\textbf{D},\textbf{D}_1$| act by zero on |$\bar{W}_{\lambda }$|⁠, while |$T_{\omega _j}\ (j\ne i)$| act by the identity operator. Identifying further |$\bar{W}_{\lambda }$| with the space |$W^{(\mathfrak{s}\mathfrak{l}_2)}_{\lambda ^{\prime}}$| constructed for |$\mathfrak{s}\mathfrak{l}_2$| instead of |$\mathfrak{g}$| (hence, the superscript in our notations), |$\textbf{D}_1$| gives rise to the operator |$\textbf{D}^{(\mathfrak{s}\mathfrak{l}_2)}_V$| with |$V$| being the restriction of the 1st fundamental |$U_v(\mathfrak{g})$|-representation |$V_1$| to the subalgebra generated by |$E_i,F_i,L^{\pm 1}_i$|⁠, which is isomorphic to |$U_{v_i}(\mathfrak{s}\mathfrak{l}_2)$|⁠. As |$V$| is not a trivial |$U_{v_i}(\mathfrak{s}\mathfrak{l}_2)$|-module, |$\textbf{D}^{(\mathfrak{s}\mathfrak{l}_2)}_V$| is a nonconstant polynomial in the 1st hamiltonian |$\textbf{D}^{(\mathfrak{s}\mathfrak{l}_2)}_1$|⁠. Hence, Lemma C.2 can be applied with |$\textbf{D}^{\prime}=\textbf{D}^{(\mathfrak{s}\mathfrak{l}_2)}_V$| and |$\textbf{D}$| denoting the image of |$\textbf{D}$| acting on |$\bar{W}_{\lambda }\simeq W^{(\mathfrak{s}\mathfrak{l}_2)}_{\lambda ^{\prime}}$| by abuse of notation. Therefore, both claims (I) and (II) follow. It is clear that |$({\mathcal{D}}^{\leq }_v(H^{\textrm{ad}})^0_{ev})^W$| is generated by |$\{\textbf{D}_i^{(0)}\}_{i=1}^n$|⁠. Hence, due to Proposition C.1, there exists a polynomial |$P$| in |$n$| variables such that |$\textbf{D}^{\prime}:=\textbf{D}-P(\textbf{D}_1,\ldots ,\textbf{D}_n)$| is of strictly negative degree. Thus, the action of |$\textbf{D}^{\prime}$| on |$W_{\lambda }$| is upper-triangular with zeros on the diagonal. As |$[\textbf{D}^{\prime},\textbf{D}_1]=0$| and |$\textbf{D}_1$| acts on |$W_{\lambda }$| with a simple spectrum, we immediately get |$\textbf{D}^{\prime}=0$|⁠. This completes our proof of Theorem 3.3. D Proof of Theorem 3.1 Given a rank |$n$| simple Lie algebra |$\mathfrak{g}$|⁠, fix an arbitrary orientation of the edges of |$\textrm{Dyn}(\mathfrak{g})$| as well as their labeling by numbers from |$1$| up to |$n-1$|⁠. For an edge |$1\leq e\leq n-1$|⁠, the vertices |$t(e),h(e)$| will denote the tail and the head of that edge, respectively. To every pair of Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$|⁠, we associate an invariant |$\vec{\epsilon }=(\epsilon _{n-1},\ldots ,\epsilon _1)\in \{-1,0,1\}^{n-1}$| via |$\epsilon _e:=\frac{\epsilon ^+_{t(e),h(e)}-\epsilon ^-_{t(e),h(e)}}{2}\in \{-1,0,1\}$| for |$1\leq e\leq n-1$|⁠. To prove Theorem 3.1, it suffices to verify that given two pairs of Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| and |$(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| satisfying |$\vec{\epsilon }=\vec{\tilde{\epsilon }}$|⁠, there exists an automorphism of |${\mathcal{D}}_v(H^{\textrm{ad}})$| that maps |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |${\mathcal{T}}(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. Our proof is similar to that of Theorem 3.2 presented in Appendix B, but is crucially based on the fermionic formula of Theorem 4.7 for |$\tilde{J}_{\beta }$| instead of Propositions 3.11, 3.14, 3.17, 3.20, and 3.38 (we owe this observation to A. Braverman). Following Appendix B, consider the action of |${\mathcal{D}}_v^{\leq }(H^{\textrm{ad}})$| on |$W_{\lambda }$|⁠. Due to Remark B.2, the action of pairwise commuting operators |$\textbf{D}_i(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| (resp. |$\textbf{D}_i(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠) is simultaneously diagonalizable in the basis of |$J$|-functions |$\{J^{\Lambda }(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm };\{y_i\})|\Lambda =\lambda -\rho -\beta ,\beta \in Q_+\}$| (resp. |$\{J^{\Lambda }(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm };\{y_i\})| \Lambda =\lambda -\rho -\beta ,\beta \in Q_+\}$|⁠) (as |$\Lambda $| varies, we will use the notations |$J^{\Lambda }(\{y_i\}), J^{\Lambda }_{\beta }$| instead of |$J(\{y_i\}), J_{\beta }$| used in Section 4). We note that both |$J^{\Lambda }(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm };\{y_i\})-\underline{y}^{\beta -\lambda }$| and |$J^{\Lambda }(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm };\{y_i\})-\underline{y}^{\beta -\lambda }$| contain only |$\{\underline{y}^{\gamma -\lambda }\}_{\gamma>\beta }$|⁠. The following is the key observation. Proposition D.1. If |$\vec{\epsilon }=\vec{\tilde{\epsilon }}$|⁠, there exists a difference operator |${\mathfrak{D}}$| that acts on |$W_{\lambda }$| and maps |$J^{\Lambda }(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm };\{y_i\})$| to a nonzero multiple of |$J^{\Lambda }(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm };\{y_i\})$| for any |$\Lambda \in \lambda -\rho -Q_+$|⁠. Proof. Define |$\nu ^{\pm }_i,\tilde{\nu }^{\pm }_i\in P$| via |$\nu ^{\pm }_i:=\sum _{k=1}^n \mathsf{n}^{\pm }_{ik}\omega _k$| and |$\tilde{\nu }^{\pm }_i:=\sum _{k=1}^n \tilde{\mathsf{n}}^{\pm }_{ik}\omega _k$|⁠. Due to Theorem 4.7, the pairing |$J^{\Lambda }_{\beta }(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| depends only on |$\{\nu ^+_i-\nu ^-_i\}_{i=1}^n$| and |$\{c^+_ic^-_i\}_{i=1}^n$| for any fixed |$\Lambda ,\beta $|⁠. Hence, as |$\vec{\epsilon }=\vec{\tilde{\epsilon }}$|⁠, we may assume |$\tilde{\epsilon }^{\pm }=\epsilon ^{\pm }, \tilde{\mathsf{n}}^-=\mathsf{n}^-,\tilde{c}^-=c^-$|⁠, while |$\gamma _i:=\tilde{\nu }^+_i-\nu ^+_i$| satisfy \begin{equation} (\alpha_i,\gamma_j)=(\alpha_j,\gamma_i)\ \textrm{for any}\ 1\leq i,j\leq n. \end{equation} (D1) In this set-up, we have the following. Lemma D.2. There exist constants |$\{s_i\}_{i=1}^n$| such that \begin{equation*} J^{\Lambda}_{\beta}(\tilde{\epsilon}^{\pm},\tilde{\mathsf{n}}^{\pm},\tilde{c}^{\pm})= v^{\sum_{i=1}^n s_i(\beta,\omega_i)+\frac{1}{2}\sum_{i=1}^n (\beta,\omega^{\vee}_i)(\beta-2\Lambda,\gamma_i)} \cdot J^{\Lambda}_{\beta}(\epsilon^{\pm},\mathsf{n}^{\pm},c^{\pm}) \end{equation*} for any |$\Lambda \in \lambda -\rho -Q_+, \beta \in Q_+$|⁠. This essentially follows from [7, (3.8, 3.9)], but let us provide a complete argument. Proof. Since |$\tilde{\epsilon }^{\pm }=\epsilon ^{\pm }, \tilde{\mathsf{n}}^-=\mathsf{n}^-,\tilde{c}^-=c^-$| and |$J^{\Lambda }_{\beta }(\bullet ,\bullet ,\bullet )$| is defined via (4.2), it suffices to prove the following equality: \begin{equation} \theta^{\Lambda}_{\beta}(\epsilon^+,\tilde{\mathsf{n}}^+,\tilde{c}^+)= \theta^{\Lambda}_{\beta}(\epsilon^+,\mathsf{n}^+,c^+)\cdot a_{\Lambda,\beta}, \end{equation} (D2) where \begin{equation} a_{\Lambda,\beta}:= v^{\frac{1}{2}\sum_{i=1}^n (\beta,\omega^{\vee}_i)(\beta-2\Lambda,\gamma_i)} \cdot \prod_{i=1}^n (\widetilde{c}^+_iv^{\frac{1}{2}(\gamma_i,\alpha_i)}/c^+_i)^{(\beta,\omega^{\vee}_i)}. \end{equation} (D3) Let |$\widetilde{\theta }^{\Lambda }_{\beta }$| denote the right-hand side of (D2). To prove (D2), it suffices to verify that |$\sum _{\beta \in Q_+} \widetilde{\theta }^{\Lambda }_{\beta }$| satisfies the defining conditions (4.1) of the Whittaker vector associated with |$(\epsilon ^+,\tilde{\mathsf{n}}^+,\tilde{c}^+)$|⁠. First, the equality |$a_{\Lambda ,0}=1$| implies |$\widetilde{\theta }^{\Lambda }_0=\theta ^{\Lambda }_0(\epsilon ^+,\mathsf{n}^+,c^+)=\textbf{1}$|⁠. Second, we note that the equality |$E_iK_{\nu ^+_i}(\theta ^{\Lambda }_{\beta +\alpha _i}(\epsilon ^+,\mathsf{n}^+,c^+))= c^+_i\cdot \theta ^{\Lambda }_{\beta }(\epsilon ^+,\mathsf{n}^+,c^+)$| implies |$E_iK_{\nu ^+_i+\gamma _i}(\widetilde{\theta }^{\Lambda }_{\beta +\alpha _i})= c^+_iv^{(\gamma _i,\Lambda -\beta -\alpha _i)}\frac{a_{\Lambda ,\beta +\alpha _i}}{a_{\Lambda ,\beta }}\cdot \widetilde{\theta }^{\Lambda }_{\beta }$|⁠. Therefore, it remains to verify \begin{equation} c^+_iv^{(\gamma_i,\Lambda-\beta-\alpha_i)}\frac{a_{\Lambda,\beta+\alpha_i}}{a_{\Lambda,\beta}}=\widetilde{c}^+_i. \end{equation} (D4) Recalling the definition of |$a_{\Lambda ,\beta }$| of (D3), we find \begin{multline*} \frac{a_{\Lambda,\beta+\alpha_i}}{a_{\Lambda,\beta}}= \frac{\widetilde{c}^+_i}{c^+_i}v^{\frac{1}{2}(\gamma_i,\alpha_i)}\cdot v^{\frac{1}{2}\sum_{j=1}^n \{(\beta+\alpha_i,\omega^{\vee}_j)(\beta-2\Lambda+\alpha_i,\gamma_j)-(\beta,\omega^{\vee}_j)(\beta-2\Lambda,\gamma_j)\}}=\\ \frac{\widetilde{c}^+_i}{c^+_i}v^{\frac{1}{2}(\gamma_i,\alpha_i)}\cdot v^{\frac{1}{2}(\beta-2\Lambda+\alpha_i,\gamma_i)+\frac{1}{2}\sum_j(\beta,\omega^{\vee}_j)(\alpha_i,\gamma_j)}= \frac{\widetilde{c}^+_i}{c^+_i}v^{(\beta-\Lambda+\alpha_i,\gamma_i)}, \end{multline*} where we used (D1) to evaluate |$\sum _j(\beta ,\omega ^{\vee }_j)(\alpha _i,\gamma _j)=\sum _j(\beta ,\omega ^{\vee }_j)(\alpha _j,\gamma _i)=(\beta ,\gamma _i)$|⁠. This implies (D4), which completes our proof of Lemma D.2. Set \begin{equation*} {\mathfrak{D}}:=\exp\left(\sum_{i=1}^n\frac{\log(T_{\omega_i})\log(T_{\gamma_i})}{2{\mathsf{d}}_i\log(v)}- \sum_{i=1}^n s_i\log(T_{\omega_i})\right). \end{equation*} This definition is motivated by the following result. Lemma D.3. |${\mathfrak{D}}(J^{\Lambda }(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm };\{y_i\}))$| is a nonzero multiple of |$J^{\Lambda }(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm };\{y_i\})$| for any |$\Lambda \in \lambda -\rho -Q_+$|⁠. Proof. Evoking formula (4.8), we get \begin{equation*} {\mathfrak{D}}(\underline{y}^{\beta-\Lambda})=\underline{y}^{\beta-\Lambda}\cdot v^{\frac{1}{2}\sum_{i=1}^n (\omega^{\vee}_i,\beta-\Lambda)(\gamma_i,\beta-\Lambda)+\sum_i s_i(\omega_i,\beta-\Lambda)}. \end{equation*} Combining this with Lemma D.2, the statement reduces to the |$\beta $|-independence of \begin{multline*} \frac{1}{2}\sum_{i=1}^n (\omega^{\vee}_i,\beta-\Lambda)(\gamma_i,\beta-\Lambda)+\sum_i s_i(\omega_i,\beta-\Lambda)- \frac{1}{2}\sum_{i=1}^n \frac{(\beta,\omega^{\vee}_i)(\beta-2\Lambda,\gamma_i)}{2}-\sum_{i=1}^n s_i(\beta,\omega_i)=\\ \sum_{i=1}^n \frac{(\omega^{\vee}_i,\beta)(\gamma_i,\Lambda)-(\omega^{\vee}_i,\Lambda)(\gamma_i,\beta-\Lambda)}{2}-\sum_{i} s_i(\omega_i,\Lambda). \end{multline*} The latter follows from |$\sum _{i=1}^n (\omega ^{\vee }_i,\beta )(\gamma _i,\Lambda )=\sum _{i=1}^n (\omega ^{\vee }_i,\Lambda )(\gamma _i,\beta )$|⁠, due to (D1). This completes our proof of Proposition D.1. Due to Proposition D.1, |${\mathfrak{D}} \textbf{D}_i(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm }) {\mathfrak{D}}^{-1}$| and |$\textbf{D}_i(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$| act diagonally in the basis |$\{J^{\Lambda }(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm };\{y_i\})|\Lambda \in \lambda -\rho -Q_+\}$| of |$W_{\lambda }$| with the same eigenvalues, hence, they coincide for every |$1\leq i\leq n$|⁠. Therefore, conjugation by |${\mathfrak{D}}$| is a well-defined automorphism of |${\mathcal{D}}_v(H^{\textrm{ad}})$| that maps |${\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |${\mathcal{T}}(\tilde{\epsilon }^{\pm },\tilde{\mathsf{n}}^{\pm },\tilde{c}^{\pm })$|⁠. This completes our proof of Theorem 3.1. E Proof of Theorem 3.24 The proof of Theorem 3.24 is similar to the one of Proposition 3.11 given in Appendix A and of Theorem 3.2 given in Appendix B, but we provide details as the formulas are different. Proof of part (a). Given a pair of type |$A_{n-1}$| Sevostyanov triples |$(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| and |$\vec{k}=(k_n,\ldots ,k_1)\in \{-1,0,1\}^n$| satisfying |$k_{i+1}=\frac{\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}}{2}$| for |$1\leq i\leq n-2$|⁠, we will prove that there exist constants |$\{r_{ij},r_i\}_{1\leq i\leq j\leq n}$| such that the function |$F$| defined in (A1) satisfies the equality \begin{equation} F^{-1}\mathsf{H}(\epsilon^{\pm},\mathsf{n}^{\pm},c^{\pm})F=\mathsf{H}_2^{\vec{k}}. \end{equation} (E1) We will view this as an equality in |${\mathcal{A}}_n$|⁠, treating |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| as an element of |${\mathcal{A}}_n$|⁠. |$\bullet $| First, we note that the terms without |${\mathsf{D}}_i$|’s are the same (and equal to |$\sum _{j=1}^n {\mathsf{w}}_j^{-2}$|⁠) both in |$F^{-1}\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })F$| and |$\mathsf{H}_2^{\vec{k}}$|⁠, independently of our choice of constants |$\{r_{ij},r_i\}$|⁠. |$\bullet $| Second, we will match the terms with |$\{\frac{{\mathsf{D}}_i}{{\mathsf{D}}_{i+1}}\}_{i=1}^{n-1}$| appearing in |$F^{-1}\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })F$| and |$\mathsf{H}_2^{\vec{k}}$|⁠. Their equality is equivalent to the following system of equations on |$\{r_{ij}\}$|⁠: \begin{equation} \frac{\mathsf{m}_{ij}-\delta_{j,i}k_i-\delta_{j,i+1}k_{i+1}}{\hbar}= \begin{cases} r_{ji}-r_{j,i+1}, &\ \textrm{if}\ 1\leq j< i\\ r_{ij}-r_{i+1,j}, &\ \textrm{if}\ i+2\leq j\leq n\\ 2r_{ii}-r_{i,i+1}, &\ \textrm{if}\ j=i\\ r_{i,i+1}-2r_{i+1,i+1}, &\ \textrm{if}\ j=i+1 \end{cases} \end{equation} (E2) and the following system of equations on |$\{r_i\}$|⁠: \begin{equation} r_i-r_{i+1}=\hbar(r_{i,i+1}-r_{ii}-r_{i+1,i+1})-\hbar^{-1}\log(b_i)+\sum_{k=1}^n(n-k+1/2)\mathsf{m}_{ik}. \end{equation} (E3) Pick any |$r_{11},r_1$|⁠. It suffices to show that (E2) admits a solution since (E3) obviously admits a unique solution with a given |$r_1$| for any choice of |$r_{ij}$|⁠. For a fixed |$r_{11}$|⁠, the constants |$\{r_{ij}\}_{1\leq i\leq j\leq n}$| satisfying the 1st, 3rd, and 4th cases of (E2) are determined uniquely. It remains to verify that they also satisfy the 2nd case of (E2). We prove this by induction in |$j-i\geq 2$|⁠. (a) If |$j=i+2$|⁠, then |$r_{i,i+2}-r_{i+1,i+2}=(r_{i,i+1}-2r_{i+1,i+1})+(2r_{i+1,i+1}-r_{i+1,i+2})-(r_{i,i+1}-r_{i,i+2})= \hbar ^{-1}(\mathsf{m}_{i,i+1}-k_{i+1}+\mathsf{m}_{i+1,i+1}-k_{i+1}-\mathsf{m}_{i+1,i})$|⁠. Hence, it remains to prove |$(\mathsf{m}_{i,i+1}-\mathsf{m}_{i,i+2})-(\mathsf{m}_{i+1,i}-\mathsf{m}_{i+1,i+1})=2k_{i+1}$|⁠. The left-hand side is equal to |$\mathsf{n}^-_{i,i+1}-\mathsf{n}^+_{i,i+1}-\mathsf{n}^-_{i+1,i}+\mathsf{n}^+_{i+1,i}= b_{i,i+1}(\epsilon ^-_{i,i+1}-\epsilon ^+_{i,i+1})=\epsilon ^+_{i,i+1}-\epsilon ^-_{i,i+1}=2k_{i+1}$|⁠, due to the choice of |$k_{i+1}$|⁠. (b) If |$j>i+2$|⁠, then |$r_{ij}-r_{i+1,j}=(r_{i,j-1}-r_{i+1,j-1})+(r_{i+1,j-1}-r_{i+1,j})-(r_{i,j-1}-r_{ij})= \hbar ^{-1}(\mathsf{m}_{i,j-1}+\mathsf{m}_{j-1,i+1}-\mathsf{m}_{j-1,i})$|⁠. Hence, it remains to prove |$(\mathsf{m}_{i,j-1}-\mathsf{m}_{ij})-(\mathsf{m}_{j-1,i}-\mathsf{m}_{j-1,i+1})=0$|⁠. The left-hand side is equal to |$b_{i,j-1}(\epsilon ^-_{i,j-1}-\epsilon ^+_{i,j-1})=0$| as |$b_{i,j-1}=0$|⁠. Thus, we have determined a collection of constants |$\{r_{ij},r_i\}_{1\leq i\leq j\leq n}$| satisfying (E2, E3). |$\bullet $| Finally, it remains to verify that for |$F$| of (A1) with the constants |$r_{ij},r_i$| chosen as above the terms with |$\frac{{\mathsf{D}}_i}{{\mathsf{D}}_j}\ (j>i+1)$| in |$F^{-1}\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })F$| and |$\mathsf{H}_2^{\vec{k}}$| do coincide. First, we note that the conditions |$\epsilon ^{\pm }_{i,i+1}=\ldots =\epsilon ^{\pm }_{j-2,j-1}=\pm 1$| and |$k_{i+1}=\ldots =k_{j-1}=1$| are equivalent under our assumption. Pick |$j>i+1$| such that either of these equivalent conditions is satisfied. Then the compatibility of the terms with |$\frac{{\mathsf{D}}_i}{{\mathsf{D}}_j}$| is equivalent to the following equality: \begin{multline} \frac{F({\mathsf{w}}_1,\ldots, v{\mathsf{w}}_i,\ldots,v^{-1}{\mathsf{w}}_j,\ldots,{\mathsf{w}}_n)}{F({\mathsf{w}}_1,\ldots,{\mathsf{w}}_n)}=\\ \prod_{k=1}^n {\mathsf{w}}_k^{\sum_{s=i}^{j-1}\mathsf{m}_{sk}+\delta_{k,i}+\delta_{k,j}}\cdot \prod_{p=i}^j {\mathsf{w}}_p^{-k_p-1}\cdot \prod_{s=i}^{j-1}b_s^{-1}\cdot v^{i+1-j+\sum_{i\leq a<b\leq j-1} (\mathsf{n}^-_{ab}-\mathsf{n}^+_{ab})+\sum_{k=1}^n \sum_{s=i}^{j-1} \frac{n+1-2k}{2}\mathsf{m}_{sk}}. \end{multline} (E4) This equality is proved by induction in |$j-i$|⁠, factoring the left-hand side as in (A6) and noticing that the |$j=i+1$| counterpart of (E4) is just the compatibility of the terms with |$\frac{{\mathsf{D}}_i}{{\mathsf{D}}_{i+1}}$|⁠, established in the previous step. Thus, the function |$F$| defined via (A1) with the constants |$\{r_{ij},r_i\}_{1\leq i\leq j\leq n}$| determined in our 2nd step satisfies the equality (E1). This completes our proof of Theorem 3.24(a). Proof of part (b). Let us write \begin{equation*} {\mathsf{T}}^v_{\vec{k}}(z)_{11}= (-1)^n{\mathsf{w}}_1\cdots{\mathsf{w}}_nz^s\left(1-\mathsf{H}^{\vec{k}}_2 z+\mathsf{H}^{\vec{k}}_3 z^2-\ldots+(-1)^n\mathsf{H}^{\vec{k}}_{n+1}z^n\right), \end{equation*} where |$s=\sum _{j=1}^n \frac{k_j-1}{2}$|⁠. For |$1\leq r\leq n$|⁠, let |$\bar{\mathsf{H}}^{\vec{k}}_{r+1}\in{\mathcal{T}}^{\vec{k}}$| be the image of |$\mathsf{H}^{\vec{k}}_{r+1}$| in |$\bar{{\mathcal{A}}}_n$|⁠. Consider the summands in |$\bar{\mathsf{H}}^{\vec{k}}_{r+1}$| without |${\mathsf{D}}_i$|’s and let |$\bar{\mathsf{H}}^{\vec{k};0}_{r+1}$| denote their sum. Tracing back the definition of |${\mathsf{T}}^v_{\vec{k}}(z)$|⁠, we get |$\bar{\mathsf{H}}^{\vec{k};0}_{r+1}=\sigma _r(\{{\mathsf{w}}_j^{-2}\})$|⁠: the |$r$|-th elementary symmetric polynomial of |$\{{\mathsf{w}}_j^{-2}\}_{j=1}^n$|⁠. Thus, the image of |$\bar{\mathsf{H}}^{\vec{k}}_{r+1}$| under the anti-isomorphism |$\bar{{\mathcal{A}}}_n\to{\mathcal{D}}_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{l}_n})$| of Section 3.9 is an element of |${\mathcal{D}}^{\leq }_v(H^{\textrm{ad}}_{\mathfrak{s}\mathfrak{l}_n})$| whose action on |$W_{\lambda }$| (see Appendix B) is upper-triangular with the same diagonal matrix coefficients as in the action of |$\textbf{D}_r\in{\mathcal{T}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$|⁠. Thus, the argument of Proposition B.1 can be applied to show that the function |$F$| of part (a), which conjugates |$\mathsf{H}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| into |$\bar{\mathsf{H}}_2^{\vec{k}}$| also conjugates the preimage of |$\textbf{D}_r$| in |$\bar{{\mathcal{A}}}_n$| into |$\bar{\mathsf{H}}^{\vec{k}}_{r+1}$| for all |$1\leq r\leq n$|⁠. Therefore, conjugation with |$F$| is an automorphism of |$\bar{{\mathcal{A}}}_n$| that maps |$\widetilde{{\mathcal{T}}}(\epsilon ^{\pm },\mathsf{n}^{\pm },c^{\pm })$| to |${\mathcal{T}}^{\vec{k}}$|⁠. This completes our proof of Theorem 3.24. Funding This work was supported by the Russian Academic Excellence Project ‘5-100’ [to R.G.] and National Science Foundation [DMS–1502497 and DMS–1821185 to A.T.]. Acknowledgments We are deeply grateful to P. Etingof for suggesting this problem, stimulating discussions, and proposing the elegant argument used in the proofs of Theorems 3.2 and 3.3; to A. Braverman for the insight that our proof of Theorem 3.2 combined with Theorem 4.7 provides an immediate proof of Theorem 3.1; to B. Feigin for enlightening discussions on the Whittaker vectors and fermionic formulas; to P. Di Francesco, R. Kedem, A. Sevostyanov, and M. Yakimov for a useful correspondence; and to M. Bershtein and A. Marshakov for their interest in this work. We also thank the anonymous referee for useful suggestions that improved the exposition. Last, but not least, we are extremely grateful to M. Finkelberg for his encouragement, numerous discussions, and constant support. This paper would not appear without his interest in the current project and inspiring discussions the authors had with him. We would like to use this opportunity to express our gratitude to M. Finkelberg for all the beautiful mathematics he has taught us for more than a decade. 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