Macdonald Duality and the proof of the Quantum Q-system conjecture

Abstract

The ${\textsf{SL}}(2,{{\mathbb {Z}}})$-symmetry of Cherednik’s spherical double affine Hecke algebras in Macdonald theory includes a distinguished generator which acts as a discrete time evolution of Macdonald operators, which can also be interpreted as a torus Dehn twist in type A. We prove for all twisted and untwisted affine algebras of type ABCD that the time-evolved q-difference Macdonald operators, in the $t\rightarrow \infty$ q-Whittaker limit, form a representation of the associated discrete integrable quantum Q-systems, which are obtained, in all but one case, via the canonical quantization of suitable cluster algebras. The proof relies strongly on the duality property of Macdonald and Koornwinder polynomials, which allows, in the q-Whittaker limit, for a unified description of the quantum Q-system variables and the conserved quantities as limits of the time-evolved Macdonald operators and the Pieri operators, respectively. The latter are identified with relativistic q-difference Toda Hamiltonians. A crucial ingredient in the proof is the use of the “Fourier transformed” picture, in which we compute time-translation operators and prove that they commute with the Pieri operators or Hamiltonians. We also discuss the universal solutions of Koornwinder-Macdonald eigenvalue and Pieri equations, for which we prove a duality relation, which simplifies the proofs further.

Appendices

Appendix A: Derivation of the ${{\mathfrak {g}}}$-Macdonald operators

We combine several constructions [40, 43, 50] of commuting difference operators corresponding to the affine algebras in Table 1. The goal is to construct an appropriate set of N commuting operators for each ${{\mathfrak {g}}}$, with eigenvalues proportional to the symmetric functions in Table 2, which form a basis for the space spanned by the irreducible fundamental characters of the Lie algebras R. This choice of ${{\mathfrak {g}}}$-Macdonald difference operators is designed such that their q-Whittaker limits satisfy the type ${{\mathfrak {g}}}$ quantum Q-systems.

1.1 Macdonald’s operators

For each affine algebra ${{\mathfrak {g}}}$ in Table 1, except for the case of $A_{2n}^{(2)}$, and for each minuscule co-weight of S, Macdonald defines a difference operator with eigenvalue which is a fundamental character of $R^*$ [40].

Let $\{e_i\}_{1\le i\le N}$ be the standard basis of ${{\mathbb {R}}}^N$ with the standard inner product $(\cdot ,\cdot )$. For the set of variables $x=(x_1,...,x_N)$, we denote $x^v=x_1^{v_1}\cdots x_N^{v_N}$ for any $v=\sum _i v_i e_i$.

Table 4 Positive roots and fundamental weights of the finite dimensional algebras of types BCD

There is a surjective map ${}_*: { R}\rightarrow { S}$, ${\alpha }\mapsto {\alpha }_*={\alpha }/u_{\alpha }\in S$, for some real $u_{\alpha }$. In the case where $R\ne S$, $u_{\alpha }=\frac{({\alpha },{\alpha })}{2}$, so that $u_{\alpha }=1$ in all cases but for the short roots of type $B_N$ or the long roots of type $C_N$, in which case it is equal to $\frac{1}{2}$ or 2, respectively.

Let $\pi =\sum \pi _i e_i$, and define⁸

$$\begin{aligned} \Phi _\pi :=\prod _{\begin{array}{c} {\alpha }\in { R}_+ \\ (\pi ,{\alpha }_*)=1 \end{array}}\frac{1-t x^{\alpha }}{1-x^{\alpha }} , \qquad \Gamma _\pi =\prod _i \Gamma _i^{\pi _i}. \end{aligned}$$

(A.1)

For each minuscule weight $\pi$ of ${S}^\vee =\{ 2\frac{{\alpha }}{({\alpha },{\alpha })}, \, {\alpha }\in S\}$ , i.e. a weight such that $(\pi ,{\alpha }_*)\in \{0, \pm 1\}$ for all ${\alpha }\in { R}$, there is a Macdonald difference operator ${{\mathcal {E}}}_\pi$ which acts on functions f(x) by the symmetrization

$$\begin{aligned} {{\mathcal {E}}}_\pi f=\sum _{w\in W}w\left( \Phi _\pi \Gamma _\pi f \right) . \end{aligned}$$

(A.2)

We list below the explicit formulas for each case treated in [40]. The construction refers to the positive roots and fundamental weights for the simple Lie groups of types BCD in Table 4.

1.1.1 Macdonald operators for $D_N^{(1)}$

Here, $R=S$ is the root system of type $D_N$. There are three minuscule weights: $\omega _1=e_1$, $\omega _{N-1}$ and $\omega _N$. Equation (A.1) becomes

$$\begin{aligned} \Phi _{\omega _1}= & {} \prod _{j=2}^N \frac{1-t x_1 x_j}{1-x_1x_j}\frac{tx_1-x_j}{x_1-x_j} ,\ \ \Phi _{\omega _{N-1}}= \prod _{1\le i<j\le N-1} \frac{1-t x_i x_j}{1-x_ix_j} \prod _{i=1}^{N-1}\frac{t x_i-x_N}{x_i-x_N},\\ \Phi _{\omega _N}= & {} \prod _{1\le i<j\le N} \frac{1-t x_i x_j}{1-x_ix_j}. \end{aligned}$$

The Weyl group of $D_N$ acts on the set $(x_1,x_2,...,x_N)$ by permutations of the indices and inversions of an even number of variables. The three Macdonald operators are

$$\begin{aligned} {{\mathcal {E}}}_{\omega _1}^{(D_N^{(1)})}= & {} \sum _{\epsilon =\pm 1}\sum _{i=1}^N \prod _{j\ne i} \frac{1-t x_i^\epsilon x_j}{1-x_i^\epsilon x_j} \frac{t x_i^\epsilon -x_j}{x_i^\epsilon -x_j} \Gamma _i^{\epsilon } , \end{aligned}$$

(A.3)

$$\begin{aligned} {{\mathcal {E}}}_{\omega _{N-1}}^{(D_N^{(1)})}= & {} \sum _{\begin{array}{c} \epsilon _1,...,\epsilon _N=\pm 1\\ \epsilon _1\epsilon _2\cdots \epsilon _N=-1 \end{array}} \prod _{1\le i<j\le N} \frac{1-t x_i^{\epsilon _i} x_j^{\epsilon _j}}{1-x_i^{\epsilon _i}x_j^{\epsilon _j}} \prod _{i=1}^N \Gamma _i^{\frac{\epsilon _i}{2}}, \end{aligned}$$

(A.4)

$$\begin{aligned} {{\mathcal {E}}}_{\omega _N}^{(D_N^{(1)})}= & {} \sum _{\begin{array}{c} \epsilon _1,...,\epsilon _N=\pm 1\\ \epsilon _1\epsilon _2\cdots \epsilon _N=1 \end{array}} \prod _{1\le i<j\le N} \frac{1-t x_i^{\epsilon _i} x_j^{\epsilon _j}}{1-x_i^{\epsilon _i}x_j^{\epsilon _j}} \prod _{i=1}^N \Gamma _i^{\frac{\epsilon _i}{2}}. \end{aligned}$$

(A.5)

These will be identified below as ${{\mathcal {D}}}_a^{(D_N^{(1)})}(x;q,t)={{\mathcal {E}}}_{\omega _{a}}^{(D_N^{(1)})}$ with $a=1,N-1,N$.

1.1.2 Macdonald operator for $B_N^{(1)}$

Here, $R=S$ is the root system of $B_N$ There is a unique minuscule weight of $S^\vee =C_N$, $\pi =e_1=\omega _1$, with

$$\begin{aligned} \Phi _{\omega _1}=\frac{1-t x_1}{1-x_1}\prod _{j=2}^N \frac{1-t x_1 x_j}{1-x_1x_j}\frac{t x_1-x_j}{x_1-x_j}. \end{aligned}$$

The Weyl group $W\simeq S_N \ltimes {{\mathbb {Z}}}_2$ is generated by all permutations and inversions of the variables in the set $x=(x_1,x_2,...,x_N)$, so the corresponding difference operator is

$$\begin{aligned} {{\mathcal {E}}}_{\omega _1}^{(B_N^{(1)})}=\sum _{\epsilon =\pm 1} \sum _{i=1}^N \frac{1-t x_i^{\epsilon }}{1-x_i^{\epsilon }} \prod _{j\ne i} \frac{1-t x_i^{\epsilon } x_j}{1-x_i^{\epsilon }x_j}\frac{t x_i^{\epsilon } -x_j}{x_i^{\epsilon }-x_j} \Gamma _i^{\epsilon }. \end{aligned}$$

This will be identified as ${{\mathcal {D}}}_1^{(B_N^{(1)})}(x;q,t)={{\mathcal {E}}}_{\omega _1}^{(B_N^{(1)})}$.

1.1.3 Macdonald operator for ${{\mathfrak {g}}}=C_N^{(1)}$

Here, $R=S$ is the root system of type $C_N$. There is a unique minuscule weight of $S^\vee =B_N$, $\pi =\frac{1}{2}\sum _{i=1}^N e_i=\omega _N$, with

$$\begin{aligned} \Phi _{\omega _N}=\prod _{i=1}^N \frac{1-t x_i^2}{1-x_i^2} \prod _{1\le i<j\le N}\frac{1-t x_i x_j}{1-x_ix_j}. \end{aligned}$$

The Weyl group is is the same as for type $B_N$, resulting in the difference operator

$$\begin{aligned} {{\mathcal {E}}}_{\omega _N}^{(C_N^{(1)})}=\sum _{\epsilon _1,...,\epsilon _N=\pm 1}\prod _{i=1}^N \frac{1-t x_i^{2\epsilon _i}}{1-x_i^{2\epsilon _i}} \prod _{1\le i<j\le N} \frac{1-t x_i^{\epsilon _i}x_j^{\epsilon _j}}{1-x_i^{\epsilon _i}x_j^{\epsilon _j}} \prod _{i=1}^N \Gamma _i^{\frac{\epsilon _i}{2}}. \end{aligned}$$

(A.6)

This will be identified as ${{\mathcal {D}}}_N^{(C_N^{(1)})}(x;q,t)={{\mathcal {E}}}_{\omega _N}^{(C_N^{(1)})}$.

1.1.4 Macdonald operator for ${{\mathfrak {g}}}=A_{2N-1}^{(2)}$

Here, $(R,S)=(C_N,B_N)$. The map $*: R\rightarrow S$ is given by $(e_i\pm e_j)_*=e_i\pm e_j$, and $(2e_i)_*=e_i$. There is a unique minuscule weight of $S^\vee =C_N$, $\pi =e_1=\omega _1$, and

$$\begin{aligned} \Phi _{\omega _1}=\frac{1-t x_1^2}{1-x_1^2} \prod _{j=2}^{N}\frac{1-t x_1 x_j}{1-x_1x_j}\frac{t x_1-x_j}{x_1-x_j}. \end{aligned}$$

Summing over the Weyl group of type $C_N$,

$$\begin{aligned} {{\mathcal {E}}}_{\omega _1}^{(A_{2N-1}^{(2)})}=\sum _{\epsilon =\pm 1} \sum _{i=1}^N \frac{1-t x_i^{2\epsilon }}{1-x_i^{2\epsilon }} \prod _{j\ne i} \frac{1-t x_i^{\epsilon } x_j}{1-x_i^{\epsilon }x_j}\frac{t x_i^{\epsilon } -x_j}{x_i^{\epsilon }-x_j} \Gamma _i^{\epsilon }. \end{aligned}$$

This will be identified as ${{\mathcal {D}}}_1^{(A_{2N-1}^{(2)})}(x;q,t)={{\mathcal {E}}}_{\omega _1}^{(A_{2N-1}^{(2)})}$.

Remark A.1

The algebra $A_{2N-1}^{(2)}$ is obtained from $A_{2N-1}^{(1)}$ by a folding procedure using the natural ${{\mathbb {Z}}}_2$ automorphism. Remarkably, this extends to the difference operators as follows. Consider the specialization $\tau$ of $x=(x_1,x_2,...,x_{2N})$ obtained by setting $x_{2N+1-i}=x_i^{-1}$, $i=1,2,...,N$, and accordingly $\Gamma _{2N+1-i}=\Gamma _i^{-1}$. We have

$$\begin{aligned} \tau \big ({{\mathcal {D}}}_1^{(A_{2N-1}^{(1)})}(x;q,t)\big )={{\mathcal {D}}}_1^{(A_{2N-1}^{(2)})}(x;q,t). \end{aligned}$$

However, the $A_{2N-1}^{(1)}$-Macdonald polynomials specialized via $\tau$ have a non-trivial decomposition onto the basis of $A_{2N-1}^{(2)}$-Macdonald polynomials.

1.1.5 Macdonald operator for ${{\mathfrak {g}}}=D_{N+1}^{(2)}$

Here, $(R,S) = (B_N,C_N)$. The map $*: R\rightarrow S$ is given by $(e_i\pm e_j)_*=e_i\pm e_j$, and $(e_i)_*=2e_i$. There is a unique minuscule weight $\pi =\frac{1}{2}\sum _{i=1}^N e_i=\omega _N$ of type $S^\vee =B_N$, so that

$$\begin{aligned} \Phi _{\omega _N}=\prod _{i=1}^N \frac{1-t x_i}{1-x_i} \prod _{1\le i<j\le N}\frac{1-t x_i x_j}{1-x_ix_j}. \end{aligned}$$

Summing over the Weyl group of type $C_N$ gives

$$\begin{aligned} {{\mathcal {E}}}_{\omega _N}^{(D_{N+1}^{(2)})}= \sum _{\epsilon _1,...,\epsilon _N=\pm 1} \prod _{i=1}^N \frac{1-t x_i^{\epsilon _i}}{1-x_i^{\epsilon _i}} \prod _{1\le i<j\le N} \frac{1-t x_i^{\epsilon _i} x_j^{\epsilon _j}}{1-x_i^{\epsilon _i} x_j^{\epsilon _j}} \prod _{i=1}^N \Gamma _i^{\epsilon _i/2}. \end{aligned}$$

(A.7)

This will be identified as ${{\mathcal {D}}}_N^{(D_{N+1}^{(2)})}(x;q,t)={{\mathcal {E}}}_{\omega _N}^{(D_{N+1}^{(2)})}$.

1.2 Higher order Koornwinder-Macdonald operators

For generic parameters (a, b, c, d, q, t), Koornwinder defined the first order q-difference operator whose eigenfunctions are the Koornwinder polynomials, invariant under the Weyl group of type C. Consequently, van Diejen [50] defined a commuting family of higher order difference operators with the same eigenfunctions. We recall this construction in A.2.1. Using the spectrum of these operators, we construct in  A.2.2 linear combinations of these, such that their eigenvalues are proportional to elementary symmetric functions ${{\hat{e}}}_a(s)$. We add to this certain higher order operators due to Rains [43]. Upon specialization of the parameters (a,b,c,d), we combine this in Sect. A.2.4, with Macdonald’s construction of Sect. A.1.

1.2.1 van Diejen’s higher order Koornwinder difference operators

Definition A.2

The van Diejen operator of order $m\in [1,N]$ is

$$\begin{aligned} {{\mathcal {V}}}_m^{(a,b,c,d)}:= & {} \sum _{\begin{array}{c} J\subset [1,N],\ |J|=m\\ \epsilon _j=\pm 1 \ \forall \ j\in J \end{array}}\sum _{s=1}^m (-1)^{s-1} \!\!\!\!\nonumber \\{} & {} \times \sum _{\emptyset \subsetneq J_1\subsetneq \cdots \subsetneq J_s=J} \prod _{r=1}^s V_{\{x\},\{\epsilon \};J_r \setminus J_{r-1};K_r}^{(a,b,c,d)} \left( \prod _{j\in J_1} \Gamma _j^{\epsilon _j}-1\right) , \end{aligned}$$

(A.8)

where $J_0=\emptyset$, $K_r=[1,N]\setminus J_r$ and

$$\begin{aligned} V_{\{x\},\{\epsilon \};J;K}^{(a,b,c,d)}:= & {} \prod _{i\in J} \frac{(1-a x_i^{\epsilon _i})(1- b x_i^{\epsilon _i})(1-c x_i^{\epsilon _i})(1-d x_i^{\epsilon _i})}{(1- x_i^{2\epsilon _i})(1- qx_i^{2\epsilon _i})} \nonumber \\{} & {} \times \prod _{\begin{array}{c} {i<j}\\ {i,j\in J} \end{array}}\frac{1-t x_i^{\epsilon _i}x_j^{\epsilon _j}}{1-x_i^{\epsilon _i}x_j^{\epsilon _j}}\frac{1-qt x_i^{\epsilon _i}x_j^{\epsilon _j}}{1-qx_i^{\epsilon _i}x_j^{\epsilon _j}} \prod _{\begin{array}{c} i\in J\\ j\in K \end{array}} \frac{1-t x_i^{\epsilon _i}x_j}{1-x_i^{\epsilon _i}x_j} \frac{t x_i^{\epsilon _i}-x_j}{x_i^{\epsilon _i}-x_j}, \nonumber \\ \end{aligned}$$

(A.9)

where $J,K\subset [1,N]$ are such that $J\cap K=\emptyset$.

When $m=1$, the van Diejen operator is the Koornwinder operators of Eq. (3.2).

Example A.3

Define

$$\begin{aligned} A^{(a,b,c,d)}(x)=\frac{(1-ax)(1-bx)(1-cx)(1-dx)}{(1-x^2)(1-q x^2)}. \end{aligned}$$

(A.10)

Then

$$\begin{aligned} {{\mathcal {V}}}_2^{(a,b,c,d)}= & {} \sum _{\begin{array}{c} 1\le i_1<i_2\le N\\ \epsilon _1,\epsilon _2=\pm 1 \end{array}}\prod _{\ell =1}^2\left( A^{(a,b,c,d)}(x_{i_\ell }^{\epsilon _\ell }) \prod _{k\ne i_1,i_2} \frac{1-t x_{i_\ell }^{\epsilon _\ell }x_k}{1-x_{i_\ell }^{\epsilon _\ell }x_k} \frac{t x_{i_\ell }^{\epsilon _\ell }-x_k}{x_{i_\ell }^{\epsilon _\ell }-x_k}\right) \\{} & {} \times \left\{ \frac{1-t x_{i_1}^{\epsilon _1}x_{i_2}^{\epsilon _2}}{1-x_{i_1}^{\epsilon _1}x_{i_2}^{\epsilon _2}}\frac{1-q t x_{i_1}^{\epsilon _1}x_{i_2}^{\epsilon _2}}{1-q x_{i_1}^{\epsilon _1}x_{i_2}^{\epsilon _2}} (\Gamma _{i_1}^{\epsilon _1}\Gamma _{i_2}^{\epsilon _2}-1) \right. \\{} & {} - \frac{1-t x_{i_1}^{\epsilon _1}x_{i_2}}{1-x_{i_1}^{\epsilon _1}x_{i_2}} \frac{t x_{i_1}^{\epsilon _1}-x_{i_2}}{x_{i_1}^{\epsilon _1}-x_{i_2}}(\Gamma _{i_1}^{\epsilon _1}-1)\\{} & {} \left. - \frac{1-t x_{i_2}^{\epsilon _2}x_{i_1}}{1-x_{i_2}^{\epsilon _2}x_{i_1}} \frac{t x_{i_2}^{\epsilon _2}-x_{i_1}}{x_{i_2}^{\epsilon _2}-x_{i_1}} (\Gamma _{i_2}^{\epsilon _2}-1) \right\} , \end{aligned}$$

where the sum over s in (A.8) decomposes into three terms, with $s=1, J_1=J=\{i_1,i_2\}$, $s=2, J_1=\{i_1\}, J_2=J=\{i_1,i_2\}$ and $s=2, J_1=\{i_2\}, J_2=J=\{i_1,i_2\}$.

The operators (A.8) form a commuting family of difference operators with common eigenfunctions being the Koornwinder polynomials. We give a description of their eigenvalues. Let $\sigma =\sqrt{\frac{abcd}{q}}$. Recall the elementary and complete symmetric functions $e_k(x)=s_{1^k}(x)$ and $h_k(x)=s_{m}(x)$.

Definition A.4

For arbitrary $\lambda _1,...,\lambda _N$, and $k,m\in [1,N]$, we define the collections of variables

$$\begin{aligned} u^{(k)}:= & {} \{\sigma t^{k-i}\}_{1\le i\le k},\\ v:= & {} \{ s_i+s_i^{-1}\}_{1\le i\le N}, \\ w_{(m)}:= & {} \{ \sigma t^{N-i} +\sigma ^{-1}t^{-(N-i)} \}_{m\le i\le N}, \end{aligned}$$

with $s_i=\sigma t^{N-i} q^{\lambda _i}$ as usual, and the functions

$$\begin{aligned} d_{m}^{(k)}:= & {} \sigma ^m\,t^{m(k-\frac{m+1}{2})}\, {{\hat{e}}}_m(u^{(k)}), \end{aligned}$$

(A.11)

$$\begin{aligned} d_{\lambda ;m}:= & {} \sigma ^m\,t^{m(N-\frac{m+1}{2})}\, {{\hat{e}}}_m(s), \end{aligned}$$

(A.12)

$$\begin{aligned} f_{\lambda ;m}:= & {} \sigma ^m\,t^{m(N-\frac{m+1}{2})}\, \sum _{j=0}^m (-1)^j\, e_{m-j}(v)\, h_{j}(w_{(m)}) . \end{aligned}$$

(A.13)

The spectral theorem for van Diejen operators is

Theorem A.5

[50] The (monic) symmetric Koornwinder polynomials $P_\lambda ^{(a,b,c,d)}(x)$ satisfy

$$\begin{aligned} {{\mathcal {V}}}_m^{(a,b,c,d)}\, P_\lambda ^{(a,b,c,d)}(x) = f_{\lambda ;m}\, P_\lambda ^{(a,b,c,d)}(x) . \end{aligned}$$

(A.14)

1.2.2 Koornwinder-Macdonald operators

Using the spectral theorem A.14, we can construct linear combinations of van Diejen’s operators with eigenvalues equal to $d_{\lambda ;m}$. To do this we prove two combinatorial lemmas about symmetric functions. Given a set of variables $x=\{x_1,...,x_N\}$, we define associated set ${{{\tilde{x}}}}=\{x_i+x_i^{-1}, i\in [1,N]\}$, and if $\beta \le N$, define $x^{[\beta ]}=\{x_1,...,x_{\beta }\}$ and ${{\tilde{x}}}^{[\beta ]} = \{x_i+x_i^{-1}, i\in [1,\beta ]\}$. In particular, $x^{[0]}=\emptyset$.

Lemma A.6

For all $r\ge 0$ and $n\ge 1$,

$$\begin{aligned} \theta _{r,n}(u):=\sum _{\ell =0}^{r}\, (-1)^\ell e_{r-\ell }(u^{[n-\ell ]}) h_\ell (u^{[n-\ell +1]})=\delta _{r,0}. \end{aligned}$$

(A.15)

Proof

When $r=0$ the sum is trivially equal to 1. We consider $r>0$. We define two types of integer configurations. A fermionic configuration on $\{1,...,p\}$ with j particles, denoted by F, is a set of j distinct integers in the set [1, p]. A bosonic configuration on $\{1,...,p'\}$ with $j'$ particles, denoted by B, is a sequence of $j'$ integers in $[1,p']$ which are not necessarily distinct. We assign a weight $w_F=\prod _{i\in F} u_i$ to a fermionic configuration, and a weight $w_B=\prod _{i\in B} (-u_i)$ to a bosonic configuration. Moreover, to the pair (F, B), we assign a weight $w_{F,B}=w_F w_B$. The partition function of j fermions on [1, p] is $e_j(u^{[p]})$, and the partition function of $j'$ bosons on $[1,p']$ is $h_{j'}(-u^{[p']})=(-1)^{j'}h_{j'}(u^{[p']})$.

We define the following set of pairs of fermionic and bosonic configurations:

$$\begin{aligned} {\mathcal {S}}_{n,r}\!=\!\left\{ (F,B) \vert \, t_F\!:=\!\max (F)\le n-|B|, t_B\!:=\!\max (B) \le n-|B|+1, |F|+|B|\!=\!r\right\} \end{aligned}$$

where |X| is the cardinality of the set X, and if $|X|=0$ we define $\max (X)=0$. The identity (A.15) is an identity for the partition function:

$$\begin{aligned} Z_{n,r} := \sum _{(F,B)\in {\mathcal {S}}_{n,r}} w_{F,B} = \delta _{r,0}. \end{aligned}$$

To prove this, for any $r\ge 1$ we construct a fixed point-free involution $\Phi$ on the set ${\mathcal {S}}_{n,r}$, such that $\Phi (w_F w_B) = - w_F w_B$. If such an involution exists, the partition function for any $r>0$ vanishes:

$$\begin{aligned} \sum _{(F,B)\in {\mathcal {S}}_{n,r}} w_{(F,B)}=\frac{1}{2}\sum _{(F,B)\in {\mathcal {S}}_{n,r}} (w_{(F,B)}+w_{\Phi (F,B)})=0\ . \end{aligned}$$

The involution $\Phi$ is illustrated in Figure 2 in the language of particles, namely by considering F, B as the sets of integer coordinates of particles along the integer line: e.g. in the case of Fig. 2(a) we have $F=\{2,4,5,6\}$ and $B=\{1,1,1,3,4,4,7,7,7\}$. Let $(F,B)\in {\mathcal {S}}_{n,r}$. The map $\Phi$ acts by moving one particle between F and B, thus preserving $|F|+|B|=r$ and reversing the sign of the weight. It is defined as $\Phi (F,B) = (F',B')$, where

• (a) If $t_B>t_F$: $F'=F\cup t_B$, $B'=B\setminus t_B$. Since $|B'|=|B|-1$, $t_{F'} = t_B \le n-|B|+1 = n-|B'|$ and $t_{B'} \le t_B \le n-|B|+1 = n-|B'| < n-|B'|+1$. Therefore, $(F',B')\in {\mathcal {S}}_{n,r}$, while $w_{F',B'}=-w_{F,B}$.

• (b) If $t_B\le t_F$: $F' = F \setminus t_F$ and $B'=B\cup t_F$. Then $t_{F'} \le t_{F}-1 \le n-|B|-1 = n-|B'|$ and $t_{B'} = t_F \le n-|B| = n-|B'|+1$, so that $(F',B')\in {\mathcal {S}}_{n,r}$, and $w_{F',B'}=-w_{F,B}$.

The map $\Phi$ is clearly an involution. When $r>0$, $\Phi$ has no fixed points, since one can always move a particle. The Lemma follows. $\square$

Fig. 2

An illustration of the involution $\Phi$. Case (a) has $t_F=6<t_B=7$, hence we move the topmost rightmost bosonic particle to a fermionic particle at position $t_F'=t_B$. Case (b) has $t_F=6\ge t_B=5$, hence we move the fermionic particle to a bosonic one at position $t_{B'}=t_F$

Lemma A.7

There is an identity on symmetric functions:

$$\begin{aligned} {{\hat{e}}}_m(x)=\sum _{j=0}^m {{\hat{e}}}_{m-j}({ y}^{[N-j]}) \sum _{k=0}^{j} (-1)^k e_{j-k}({{{\tilde{x}}}}) h_{k}({{\tilde{y}}}^{[N-j+1]}). \end{aligned}$$

(A.16)

Proof

We rewrite (A.16) using the generating function ${{\hat{E}}}(z;x)$ of (3.1) as

$$\begin{aligned} {{\hat{E}}}(z;x)\vert _{z^m}=\sum _{j=0}^m {{\hat{E}}}(z;{y}^{[N-j]}) \vert _{z^{m-j}} F(z;{{{\tilde{x}}}},{{{\tilde{y}}}}^{[N-j+1]})\vert _{z^{j}}, \end{aligned}$$

(A.17)

where $f(z)\vert _{z^m}$ is the coefficient of $z^m$ in the series expansion of f(z) around 0, and

$$\begin{aligned} {{\hat{E}}}(z;{y}^{[\beta ]})=\prod _{i=1}^{\beta }(1+{{{\tilde{y}}}}_i z+z^2), \quad F(z;{{{\tilde{x}}}},{{{\tilde{y}}}}^{[\beta ]})=\frac{\prod _{i=1}^N 1+z{\tilde{x}}_i}{\prod _{i=1}^{\beta } 1+z{\tilde{y}}_i} \ . \end{aligned}$$

One can further decompose

$$\begin{aligned} \prod _{i=1}^{N-j} (1+z {{{\tilde{y}}}}_i+z^2)=(1+z^2)^{N-j} {{\hat{E}}}(\frac{z}{1+z^2};{\tilde{y}}) =\sum _{k=0}^{N-j} z^k\,(1+z^2)^{N-j-k} \,e_k({\tilde{y}}^{[N-j]}), \end{aligned}$$

and

$$\begin{aligned} \frac{\prod _{i=1}^N (1+z {{{\tilde{x}}}}_i)}{\prod _{i=1}^{N-j+1} (1+z {{{\tilde{y}}}}_i)}=\prod _{i=1}^N (1+z {{{\tilde{x}}}}_i) \sum _{\ell \ge 0} (-1)^\ell z^\ell h_\ell ({\tilde{y}}^{[N-j+1]}). \end{aligned}$$

The right hand side of (A.17) is therefore

$$\begin{aligned}{} & {} \sum _{j=0}^m \sum _{k,\ell } (1+z^2)^{N-j-k} \big \vert _{z^{m-j-k}}\, \prod _{i=1}^N(1+z {{{\tilde{x}}}}_i)\vert _{z^{j-\ell }}\, (-1)^\ell e_k({\tilde{y}}^{[N-j]}) h_\ell ({{\tilde{y}}}^{[N-j+1]})\nonumber \\{} & {} \quad =\sum _{r=0}^m \!\! \sum _{\begin{array}{c} k,\ell \ge 0\\ k+\ell =r \end{array}}\!\! \sum _{j=\ell }^{m-k}\! (1+z^2)^{N-j-k} \big \vert _{z^{m-j-k}}\, \! \! \prod _{i=1}^N\!(1+z {{{\tilde{x}}}}_i)\vert _{z^{j-\ell }} (-1)^\ell e_k({\tilde{y}}^{[N-j]}) h_\ell ({\tilde{y}}^{[N-j+1]}),\nonumber \\{} & {} \quad =\sum _{r=0}^m \!\! \sum _{j=0}^{m-r}\! (1\!+\!z^2)^{N-j-r} \big \vert _{z^{m-j-r}}\!\! \prod _{i=1}^N\!(1\!+\!z {{{\tilde{x}}}}_i)\vert _{z^{j}}\!\!\nonumber \\{} & {} \quad \times \sum _{\ell =0}^{r}\, \!(-1)^\ell \! e_{r-\ell }({\tilde{y}}^{[N-j-\ell ]}) h_\ell ({\tilde{y}}^{[N-j-\ell +1]}), \end{aligned}$$

(A.18)

where we changed variables $k\mapsto r-\ell$ and $j\mapsto j-\ell$. Finally, using Lemma A.6 for the collection $u={{{\tilde{y}}}}$ and $n=N-j$, the expression above drastically simplifies into

$$\begin{aligned}\sum _{j=0}^{m} (1+z^2)^{N-j} \big \vert _{z^{m-j}}\, \prod _{i=1}^N(1+z {{{\tilde{x}}}}_i)\vert _{z^{j}}=\prod _{i=1}^N (1+z {{{\tilde{x}}}}_i+z^2)\big \vert _{z^m} ={{\hat{e}}}_m(x),\end{aligned}$$

which implies (A.17). $\square$

Definition A.8

We define the Koornwinder-Macdonald difference operators as

$$\begin{aligned}{{\mathcal {D}}}_m^{(a,b,c,d)}(x):= \sum _{j=0}^m d_{j}^{(N-m+j)}\, {{\mathcal {V}}}_{m-j}^{(a,b,c,d)},\qquad \quad (m=1,2,...,N),\end{aligned}$$

in terms of the van Diejen operators of Sect.A.2.1, with $d_{j}^{(k)}$ as in (A.11).

Theorem A.9

The Koornwinder-Macdonald polynomials are common eigenfunctions of ${{\mathcal {D}}}_m$ with eigenvalue $d_{\lambda ;m}$ defined in (A.12):

$$\begin{aligned} {{\mathcal {D}}}_m^{(a,b,c,d)}(x;q,t)\, P_\lambda ^{(a,b,c,d)}(x)= d_{\lambda ;m}\, P_\lambda ^{(a,b,c,d)}(x) . \end{aligned}$$

(A.19)

Proof

Equation (A.19) follows from Theorem A.5 and the relation

$$\begin{aligned} d_{\lambda ;m} =\sum _{j=0}^m d_{j}^{(N-m+j)}\, f_{\lambda ;m-j} . \end{aligned}$$

(A.20)

Equation (A.20) is obtained by specializing Lemma A.7 to the variables $x_i=\sigma t^{N-i}q^{\lambda _i}$ and $y_i=\sigma t^{N-i}$, and noting that the prefactors in (A.12-A.13) amount to an overall factor of $\sigma ^m t^{m(N-\frac{m+1}{2})}$ on both sides of the equation. $\square$

1.2.3 Rains operators

The Rains operator $\widehat{{\mathcal {D}}}_N^{(a,b,c,d)}(x;q,t)$ defined in (3.7,3.9) has eigenvalues (3.10). This operator is not linearly independent of the K-M operators, as is seen from the following Lemma.

Lemma A.10

For generic (a, b, c, d) the following relation holds between the Koornwinder-Macdonald operators and the Rains operator:

$$\begin{aligned} \widehat{{\mathcal {D}}}_N^{(a,b,c,d)}={{\mathcal {D}}}_{N}^{(a,b,c,d)}+\sum _{m=1}^N (-1)^m\left( a^mb^m +\frac{c^md^m}{q^m}\right) \,t^{m(m-1)/2} \, {{\mathcal {D}}}_{N-m}^{(a,b,c,d)}. \nonumber \\ \end{aligned}$$

(A.21)

Proof

This is a consequence of the relation between the eigenvalues:

$$\begin{aligned} \hat{d}_{\lambda ;N}^{(a,b,c,d)}=d_{\lambda ,N}+\sum _{m=1}^N (-1)^m\left( a^mb^m +\frac{c^md^m}{q^m}\right) \,t^{m(m-1)/2} \,d_{\lambda ;N-m} . \end{aligned}$$

(A.22)

Using the notation $\sigma =\sqrt{abcd/q}$, $u=ab/\sigma$, $u^{-1}=c d/q\sigma$ and $s_i=\sigma q^{\lambda _i}t^{N-i}$, as well as the expressions (A.12) for $d_{\lambda ,m}$ and (3.10) for $\hat{d}_{\lambda ;N}$, Eq. (A.22) is

$$\begin{aligned}{} & {} q^{-|\lambda |} \prod _{i=1}^N (1\!-\!u s_i)(1\!-u^{-1}\! s_i)\!=\!\sigma ^N t^{N(N-1)/2}\!\!\left\{ \! \hat{e}_N(s)\!+\!\!\!\sum _{m=1}^N\! (-1)^m (u^m\!+\!u^{-m}) \hat{e}_{N-m}(s)\!\! \right\} \\{} & {} \quad =\frac{(-\sigma )^N\, t^{N(N-1)/2}}{u^N} \!\!\sum _{m=0}^{2N} \!(-1)^m u^m \hat{e}_m(s) \!\!=\!\!\frac{(-1)^N \sigma ^N\, t^{N(N-1)/2}}{u^N}\!\!\prod _{i=1}^N(1\!-\!u s_i)(1\!-\!u s_i^{-1}), \end{aligned}$$

where we have used (3.1) in the last line. The equality follows from $\prod _{i=1}^N s_i =q^{|\lambda |} \sigma ^N\, t^{N(N-1)/2}$. $\square$

1.2.4 Koornwinder-Macdonald operators and Macdonald’s operators of Section A.1

The first Koornwinder-Macdonald operator in Equation of Definition A.8 is

$$\begin{aligned} {{\mathcal {D}}}_1^{(a,b,c,d)}(x;q,t)= & {} {{\mathcal {V}}}_1^{(a,b,c,d)}(x;q,t)+\frac{1-t^N}{1-t}\left( 1+ \frac{abcd}{q}t^{N-1}\right) , \nonumber \\= & {} {{\mathcal {K}}}_1^{(a,b,c,d)}(x;q,t)+\frac{1-t^N}{1-t}\left( 1+ \frac{abcd}{q}t^{N-1}\right) . \nonumber \\ \end{aligned}$$

(A.23)

To relate these with some of the operators of Sect. A.1, we have the following Lemma.

Lemma A.11

The first order Koornwinder-Macdonald operator can be written as

$$\begin{aligned} \qquad {{\mathcal {D}}}_1^{(a,b,c,d)}(x;q,t)= \varphi ^{(a,b,c,d)}(x)+ \sum _{i=1}^N\sum _{\epsilon =\pm 1} \Phi _{i,\epsilon }^{(a,b,c,d)}(x) \Gamma _i^{\epsilon }, \end{aligned}$$

where

$$\begin{aligned} \varphi ^{(a,b,c,d)}(x)= & {} \sum _{\epsilon =\pm 1} \frac{\big (1-\frac{\epsilon }{ q^{1/2}}a\big ) \big (1-\frac{\epsilon }{q^{1/2}}b\big ) \big (1-\frac{\epsilon }{q^{1/2}}c\big )\big (1- \frac{\epsilon }{q^{1/2}}d\big )}{2(1-t)(1-q^{-1}t)}\nonumber \\{} & {} \times \left\{ \prod _{i=1}^N \frac{1-\frac{\epsilon \, t }{q^{1/2}}x_i}{1-\frac{\epsilon }{q^{1/2}}x_i}\, \frac{1-\frac{\epsilon \, t }{q^{1/2}}x_i^{-1}}{1-\frac{\epsilon }{q^{1/2}}x_i^{-1}} -t^N\right\} . \end{aligned}$$

(A.24)

Proof

We have:

$$\begin{aligned} \varphi ^{(a,b,c,d)}(x)=\frac{1-t^N}{1-t}\left( 1+ \frac{abcd}{q}t^{N-1}\right) -\sum _{i=1}^N\sum _{\epsilon =\pm 1} \Phi _{i,\epsilon }^{(a,b,c,d)}(x) . \end{aligned}$$

(A.25)

Using the simple fraction decomposition of the function

$$\begin{aligned} \theta (z):= \frac{\big (1-\frac{z}{a}\big )\big (1-\frac{z}{b}\big )\big (1-\frac{z}{c}\big )\big (1-\frac{z}{d}\big )}{\big (1-\frac{z^2}{q}\big )\big (1-\frac{z^2}{t}\big )} \, \prod _{i=1}^N\frac{1-\frac{z}{t} x_i}{1-z x_i}\, \frac{1-\frac{z}{t} x_i^{-1}}{1-z x_i^{-1}} . \end{aligned}$$

(A.26)

We find

$$\begin{aligned} \theta (z)=\frac{q}{abcd\,t^{2N-1}}+\sum _{\epsilon =\pm 1}\left\{ \sum _{i=1}^N \frac{A_{i,\epsilon }}{1-z x_i^{\epsilon }} +\frac{B_\epsilon }{1-\epsilon \frac{z}{q^{1/2}}}+\frac{C_\epsilon }{1-\epsilon \frac{z}{t^{1/2}}}\right\} , \end{aligned}$$

where

$$\begin{aligned} A_{i,\epsilon }= & {} \frac{1-t}{t^{2N}}\,\frac{\big (1-\frac{x_i^{-\epsilon }}{a}\big )\big (1-\frac{x_i^{-\epsilon }}{b}\big )\big (1-\frac{x_i^{-\epsilon }}{c}\big )\big (1-\frac{x_i^{-\epsilon }}{d}\big )}{\big (1-\frac{x_i^{-2\epsilon }}{q}\big )\big (1-x_i^{-2\epsilon }\big )} \, \prod _{j\ne i} \frac{t x_i^{\epsilon }-x_j}{x_i^{\epsilon }-x_j}\, \frac{t x_i^{\epsilon }x_j-1}{x_i^{\epsilon }x_j-1},\\= & {} \frac{q(1-t)}{abcd\, t^{2N-1}}\, \Phi _{i,\epsilon }^{(a,b,c,d)}(x) , \end{aligned}$$

and

$$\begin{aligned} B_\epsilon= & {} \frac{\big (1-\epsilon \frac{a}{q^{1/2}}\big )\big (1-\epsilon \frac{b}{q^{1/2}}\big )\big (1-\epsilon \frac{c}{q^{1/2}}\big )\big (1-\epsilon \frac{d}{q^{1/2}}\big )}{2\,abcd\, t^{2N-1}\,\big (1-\frac{t}{q}\big )}\prod _{i=1}^N \frac{1-\frac{\epsilon \, t }{q^{1/2}}x_i}{1-\frac{\epsilon }{q^{1/2}}x_i}\, \frac{1-\frac{\epsilon \, t }{q^{1/2}}x_i^{-1}}{1-\frac{\epsilon }{q^{1/2}}x_i^{-1}},\\ C_\epsilon= & {} q\frac{\big (1-\epsilon \frac{a}{t^{1/2}}\big )\big (1-\epsilon \frac{b}{t^{1/2}}\big )\big (1-\epsilon \frac{c}{t^{1/2}}\big )\big (1-\epsilon \frac{d}{t^{1/2}}\big )}{2\,abcd\, t^{N-1}\,\big (1-\frac{q}{t}\big )}. \end{aligned}$$

By definition (A.26), $\theta (0)=1$, therefore

$$\begin{aligned} \sum _{\epsilon =\pm 1}\left\{ \sum _{i=1}^N \Phi _{i,\epsilon }^{(a,b,c,d)}(x)+\frac{abcd\, t^{2N-1}}{q(1-t)}\big (B_\epsilon +C_{\epsilon }\big )\right\} =\frac{\frac{abcd}{q}\, t^{2N-1}-1}{1-t}. \nonumber \\ \end{aligned}$$

(A.27)

Using

$$\begin{aligned}{} & {} \sum _{\epsilon =\pm 1} \left\{ \frac{t}{2}\big (1-\epsilon \frac{a}{t^{1/2}}\big )\big (1-\epsilon \frac{b}{t^{1/2}}\big )\big (1-\epsilon \frac{c}{t^{1/2}}\big )\big (1-\epsilon \frac{d}{t^{1/2}}\big ) \right. \\{} & {} \quad \left. \ - \frac{q}{2}\big (1-\epsilon \frac{a}{q^{1/2}}\big )\big (1-\epsilon \frac{b}{q^{1/2}}\big )\big (1-\epsilon \frac{c}{q^{1/2}}\big )\big (1-\epsilon \frac{d}{q^{1/2}}\big )\right\} =\left( 1-\frac{q}{t}\right) \left( t-\frac{abcd}{q}\right) , \end{aligned}$$

we have:

$$\begin{aligned} \sum _{\epsilon =\pm 1}\frac{abcd\, t^{2N-1}}{q(1-t)}\,C_{\epsilon }= & {} \frac{t^N}{2(1-t)\big (1-\frac{q}{t}\big )} \sum _{\epsilon =\pm 1}\prod _{u=a,b,c,d} \big (1-\epsilon \frac{u}{t^{1/2}}\big )\\= & {} \frac{t^N- \frac{abcd}{q}t^{N-1}}{1-t}-\frac{t^N}{2(1-t)\big (1-\frac{t}{q}\big )}\sum _{\epsilon =\pm 1} \prod _{u=a,b,c,d}\bigg (1-\epsilon \frac{u}{q^{1/2}}\bigg ). \end{aligned}$$

The Lemma follows by substituting this into (A.27), and using the result to reexpress $\varphi ^{(a,b,c,d)}(x)$ as given by (A.25). $\square$

When $c=q^{1/2}$ and $d=-q^{1/2}$, $\varphi ^{(a,b,q^{1/2},-q^{1/2})}(x)=0$. Using Table 1, we obtain the following.

Corollary A.12

In the cases ${{\mathfrak {g}}}=D_N^{(1)},B_N^{(1)}, A_{2N-1}^{(2)}$ the first Macdonald operator is equal to the first specialized Koornwinder-Macdonald operator:

$$\begin{aligned} {{\mathcal {D}}}_1^{({{\mathfrak {g}}})}(x;q,t)=\sum _{i=1}^N\sum _{\epsilon =\pm 1} \Phi _{i,\epsilon }^{({{\mathfrak {g}}})}(x) \Gamma _i^{\epsilon }={{\mathcal {E}}}_1^{({{\mathfrak {g}}})}(x;q,t). \end{aligned}$$

(A.28)

By direct inspection, the specialized Rains operators of (3.7) and (3.9) can also be expressed in terms Macdonald’s operators.

Lemma A.13

In the cases ${{\mathfrak {g}}}=D_{N}^{(1)}, C_N^{(1)}, D_{N+1}^{(2)}$ we have the identifications

$$\begin{aligned} \begin{array}{ll} D_{N}^{(1)}:&{}{{\mathcal {E}}}_{\omega _N}^{(D_N^{(1)})}(x;q,t) +{{\mathcal {E}}}_{\omega _{N-1}}^{(D_N^{(1)})}(x;q,t)= {{\mathcal {R}}}_N^{(1,-1)}(x;q,t),\\ {} &{} \widehat{{\mathcal {D}}}_N^{(1,-1,q^{1/2},-q^{1/2})}=\left( {{\mathcal {R}}}_N^{(1,-1)}\right) ^2;\\ C_N^{(1)}:&{}{{\mathcal {E}}}_{\omega _N}^{(C_N^{(1)})}(x;q,t)={{\mathcal {R}}}_N^{(t^{1/2},-t^{1/2})}(x;q,t),\\ &{}\widehat{{\mathcal {D}}}_N^{(t^{1/2},-t^{1/2},t^{1/2}q^{1/2},-t^{1/2}q^{1/2})}=\left( {{\mathcal {R}}}_N^{(t^{1/2},-t^{1/2})}\right) ^2;\\ D_{N+1}^{(2)}:&{}{{\mathcal {E}}}_{\omega _N}^{(D_{N+1}^{(2)})}(x;q,t)={{\mathcal {R}}}_N^{(t,-1)}(x;q,t),\\ &{}\widehat{{\mathcal {D}}}_N^{(t,-1,t q^{1/2},-q^{1/2})}=\left( {{\mathcal {R}}}_N^{(t,-1)}\right) ^2. \end{array} \end{aligned}$$

(A.29)

1.3 ${{\mathfrak {g}}}$-Macdonald operators and their eigenvalues

For each ${{\mathfrak {g}}}$, we have chosen a list of N linearly independent commuting difference operators (see Def. 3.5) using the operators in the preceding sections of this Appendix. These have the property that in the q-Whittaker limit, they and their time-translates provide solutions of the various quantum Q-systems. We first explore redundancies between the various definitions to justify Def. 3.5. Then we describe the eigenvalues of the ${{\mathfrak {g}}}$-Macdonald operators, and conclude with a remark on the choice leading to Def. 3.5.

1.3.1 Redundancies

Specializing the parameters (a, b, c, d) as in Table 1 in the difference operators of Theorem A.9 gives a list of N commuting difference operators for each ${{\mathfrak {g}}}$. As noted in Corollary A.12, some of these are Macdonald’s operators listed in Sect. A.1. In addition, there are non-linear relations⁹:

$$\begin{aligned} {{\mathcal {E}}}_{\omega _N}^{(D_N^{(1)})}\,{{\mathcal {E}}}_{\omega _{N-1}}^{(D_N^{(1)})}= & {} \sum _{{\alpha }=0}^{\lfloor \frac{N-1}{2} \rfloor } t^{{\alpha }(2{\alpha }+1)}\, {{\mathcal {D}}}^{(a,b,c,d)}_{N-2{\alpha }-1},\\ ({{\mathcal {E}}}_{\omega _N}^{(D_N^{(1)})})^2+({{\mathcal {E}}}_{\omega _{N-1}}^{(D_N^{(1)})})^2= & {} {{\mathcal {D}}}_{N}^{(a,b,c,d)} +2\sum _{{\alpha }=0}^{\lfloor \frac{N}{2} \rfloor } t^{{\alpha }(2{\alpha }-1)}\, {{\mathcal {D}}}^{(a,b,c,d)}_{N-2{\alpha }}, \end{aligned}$$

for the $D_N^{(1)}$ specialization. Similarly, for the $C_N^{(1)}$ and $D_{N+1}^{(2)}$ specializations,

$$\begin{aligned} \left( {{\mathcal {E}}}_{\omega _N}^{({{\mathfrak {g}}})}\right) ^2 ={{\mathcal {D}}}_N^{(a,b,c,d)}+2\sum _{{\alpha }=1}^N t^{\frac{{\alpha }({\alpha }+1)}{2}} \, {{\mathcal {D}}}^{(a,b,c,d)}_{N-{\alpha }}. \end{aligned}$$

In those cases, the Macdonald operators carry more information than the specialization of the Koornwinder-Macdonald operators ${{\mathcal {D}}}_m^{(a,b,c,d)}$. This justifies the choice in Def. 3.5.

1.3.2 The eigenvalues of ${{\mathfrak {g}}}$-Macdonald operators

Definition A.14

If $m\le N_{{\mathfrak {g}}}$, let $d_{\lambda ,m}^{({{\mathfrak {g}}})}$ to be the specialization of Eq. A.12 to the parameters corresponding to ${{\mathfrak {g}}}$ in Table 1:

$$\begin{aligned} d_{\lambda ,m }^{({{\mathfrak {g}}})}:=d_{\lambda ,m}=t^{m(N+\xi _{{\mathfrak {g}}}-\frac{m+1}{2})}\, {\hat{e}}_m(s), \quad m\in [1,N_{{\mathfrak {g}}}], \end{aligned}$$

(A.30)

where $s=t^{\rho ^{({{\mathfrak {g}}})}} q^\lambda$, i.e. $s_i=q^{\lambda _i} t^{N-i+\xi _{{\mathfrak {g}}}}$. Otherwise, define

$$\begin{aligned} \begin{array}{ll} {{\mathfrak {g}}}=D_N^{(1)}:&{} d_{\lambda ;\beta }^{(D_{N}^{(1)})}=t^{\frac{N(N-1)}{4}}\,{{\hat{e}}}_\beta ^{(D_{N})}(s), \qquad \beta =N-1,N ,\\ {{\mathfrak {g}}}=C_N^{(1)}, D_{N+1}^{(2)}:&{} d_{\lambda ;N}^{({{\mathfrak {g}}})}=t^{\frac{N(N+1)}{4}}\,{{\hat{e}}}_N^{(B_N)}(s), \end{array} \end{aligned}$$

(A.31)

where

$$\begin{aligned} {{\hat{e}}}_\beta ^{(D_{N})}(x):= & {} \sum _{\begin{array}{c} \epsilon _1,...,\epsilon _N=\pm 1 \\ \prod \epsilon _i=2(\beta -N)+1 \end{array}} \prod _{i=1}^N x_i^{\epsilon _i/2}\qquad (\beta =N-1,N),\nonumber \\ {{\hat{e}}}_N^{(B_N)}(x):= & {} \prod _{i=1}^N \frac{1+x_i}{\sqrt{x_i}}={{\hat{e}}}_{N-1}^{(D_{N})}(x)+{{\hat{e}}}_N^{(D_{N})}(x). \end{aligned}$$

(A.32)

The functions ${\hat{e}}_m^{(R^*)}(s)$ have the property that their dominant monomial (in powers of t) is $s^{\omega _m^*}$, where $\omega _m^*$ are the fundamental weights of $R^*$.

Theorem A.15

The ${{\mathfrak {g}}}$-specialized Koornwinder polynomials are common eigenvectors of the ${{\mathfrak {g}}}$-Macdonald operators, satisfying the eigenvalue equation

$$\begin{aligned} {{\mathcal {D}}}^{({{\mathfrak {g}}})}_m(x)\, P_\lambda ^{({{\mathfrak {g}}})}= d_{\lambda ;m}^{({{\mathfrak {g}}})}\, P_\lambda ^{({{\mathfrak {g}}})} , \qquad m\in [1,N], \end{aligned}$$

(A.33)

with $d_{\lambda ;m}^{({{\mathfrak {g}}})}=\theta _m^{({{\mathfrak {g}}})}\, \hat{e}_m^{(R^*)}(s)$ ($\hat{e}_m^{(R^*)}= \hat{e}_m$ for $m\le N_{{\mathfrak {g}}}$) and

$$\begin{aligned}{} & {} \theta _m^{({{\mathfrak {g}}})}=t^{m(N+\xi _{{\mathfrak {g}}}-\frac{m+1}{2})},\qquad m\le N_{{\mathfrak {g}}}, \end{aligned}$$

(A.34)

$$\begin{aligned}{} & {} \begin{array}{c} \theta _\beta ^{(D_{N}^{(1)})}=t^{\frac{N(N-1)}{4}}, \qquad \beta =N-1,N,\\ \theta _N^{({{\mathfrak {g}}})}=t^{\frac{N(N+1)}{4}}, \qquad {{\mathfrak {g}}}=C_N^{(1)}, D_{N+1}^{(2)}. \end{array} \end{aligned}$$

(A.35)

1.3.3 Remark about our choice of Macdonald operators

We made the choice of ${{\mathfrak {g}}}$-Macdonald operators for simplicity. A more natural choice is to choose eigenvalues proportional to the fundamental characters of $R^*$. This leads to a more complicated choice of the difference operators, but in the q-Whittaker limit, they have the same limit as our ${{\mathfrak {g}}}$-Macdonald operators.

For Macdonald’s operators in Sect. A.1, the eigenvalue equation is

$$\begin{aligned} {{\mathcal {D}}}_m^{({{\mathfrak {g}}})}(x)\, P_\lambda ^{({{\mathfrak {g}}})}(x)= \theta _m^{({{\mathfrak {g}}})}\,s_{\omega _m}^{(R^*)}(s)\, P_\lambda ^{({{\mathfrak {g}}})}(x) , \end{aligned}$$

where the Schur functions $s_{\omega _m}^{(R^*)}$ are the fundamental characters of $R^*$ with highest weight $\omega _m^*$. We may choose the set of difference operators $\tilde{{\mathcal {D}}}_m^{({{\mathfrak {g}}})}(x)$, with eigenvalues proportional to the fundamental characters of $R^*$ for all m as above. These are related to ${{\mathcal {D}}}_m^{({{\mathfrak {g}}})}$ via a triangular change of basis, as can be seen from the relation between the fundamental characters and $e_m^{(R)}$:

$$\begin{aligned} D_N: \quad s_{\omega _m}^{(D_N)}= & {} \hat{e}_m^{(D_N)} ,\qquad m\in [1,N],\\ B_N: \quad s_{\omega _m}^{(C_N)}= & {} \hat{e}_m-\hat{e}_{m-2} , \qquad m\in [1,N],\\ C_N: \quad s_{\omega _m}^{(B_N)}= & {} \hat{e}_m+\hat{e}_{m-1} \qquad m\in [1,N-1],\\ s_{\omega _N}^{(B_N)}= & {} \hat{e}_N^{(B_N)}. \end{aligned}$$

Therefore $\tilde{{\mathcal {D}}}^{({{\mathfrak {g}}})}$ are

$$\begin{aligned} {{\mathfrak {g}}}=D_N^{(1)}: \quad \tilde{{\mathcal {D}}}^{({{\mathfrak {g}}})}_m:= & {} {{\mathcal {D}}}^{({{\mathfrak {g}}})}_m \qquad (m\in [1,N]),\nonumber \\ {{\mathfrak {g}}}=B_N^{(1)},A_{2N-1}^{(2)},A_{2N}^{(2)}: \quad \tilde{{\mathcal {D}}}^{({{\mathfrak {g}}})}_m:= & {} {{\mathcal {D}}}^{({{\mathfrak {g}}})}_m -(1-\delta _{m,1})\frac{\theta _m^{({{\mathfrak {g}}})}}{\theta _{m-2}^{({{\mathfrak {g}}})}}\, {{\mathcal {D}}}^{({{\mathfrak {g}}})}_{m-2} \ \ (m\in [1,N]),\nonumber \\ {{\mathfrak {g}}}=C_N^{(1)},D_{N+1}^{(2)}: \quad \tilde{{\mathcal {D}}}^{({{\mathfrak {g}}})}_m:= & {} {{\mathcal {D}}}^{({{\mathfrak {g}}})}_m +\frac{\theta _m^{({{\mathfrak {g}}})}}{\theta _{m-1}^{({{\mathfrak {g}}})}}\, {{\mathcal {D}}}^{({{\mathfrak {g}}})}_{m-1}\quad (m\in [1,N-1]),\nonumber \\ \tilde{{\mathcal {D}}}^{({{\mathfrak {g}}})}_N:= & {} {{\mathcal {D}}}^{({{\mathfrak {g}}})}_N , \end{aligned}$$

(A.36)

with the convention that ${{\mathcal {D}}}^{({{\mathfrak {g}}})}_0=1$. This choice guarantees the following:

Theorem A.16

For all $m=1,2,...,N$ and all ${{\mathfrak {g}}}$,

$$\begin{aligned} \tilde{{\mathcal {D}}}^{({{\mathfrak {g}}})}_m\, P_\lambda (x)=\theta _m^{({{\mathfrak {g}}})}\, s_{\omega _m}^{(R^*)}(s)\, P_\lambda (x), \qquad s=q^\lambda \,t^{\rho ^{({{\mathfrak {g}}})}}. \end{aligned}$$

(A.37)

1.4 q-Whittaker limit of the ${{\mathfrak {g}}}$-Macdonald operators

The q-Whittaker limit corresponds to sending $t\rightarrow \infty$. The quantity $A^{(a,b,c,d)}(x)$ of (A.10), which appears in the van Diejen operators (A.8), tends to $\sigma = t^{\xi _{{\mathfrak {g}}}}$ under the specialization of Table 1. All terms in (A.8) have the same leading behavior as that with $s=1$ and $J=J_1=\{1,2,...,m\}$, namely a sum of three contributions from (A.9):

$$\begin{aligned} m \xi _{{\mathfrak {g}}}+ m(m-1) + 2 m(N-m)=m(2N-m-1+\xi _{{\mathfrak {g}}}). \end{aligned}$$

Therefore, ${{\mathcal {V}}}_m^{({{\mathfrak {g}}})} \sim (\theta _m^{({{\mathfrak {g}}})})^2\, V_m^{({{\mathfrak {g}}})}$, where the difference operator $V_m^{({{\mathfrak {g}}})}$ is independent of t. Similarly ${{\mathcal {D}}}_m^{({{\mathfrak {g}}})}\sim (\theta _m^{({{\mathfrak {g}}})})^2\, D_m^{({{\mathfrak {g}}})}$ where $D_m^{({{\mathfrak {g}}})}$ independent of t, for $m=1,2,...,N_{{\mathfrak {g}}}$. By inspection, we find the same leading behavior for $N_{{\mathfrak {g}}}<m\le N$, using (A.35). This leads to the definitions:

$$\begin{aligned} D_m^{({{\mathfrak {g}}})}(x):=\lim _{t\rightarrow \infty } (\theta _m^{({{\mathfrak {g}}})})^{-2} \,{{\mathcal {D}}}_m^{({{\mathfrak {g}}})}(x),\ \ V_m^{({{\mathfrak {g}}})}:=\lim _{t\rightarrow \infty } (\theta _m^{({{\mathfrak {g}}})})^{-2}\, {{\mathcal {V}}}_m^{({{\mathfrak {g}}})} \end{aligned}$$

(A.38)

with $\theta _m^{({{\mathfrak {g}}})}$ as in (A.34-A.35).

Remark A.17

The limit $t\rightarrow \infty$ of ${{\mathcal {D}}}^{({{\mathfrak {g}}})}_m$ is the same as that of $\tilde{{\mathcal {D}}}^{({{\mathfrak {g}}})}_m$ of Sect. A.3.3: Using ${{\mathcal {D}}}^{({{\mathfrak {g}}})}_m\simeq (\theta _m^{({{\mathfrak {g}}})})^{2}\, {D}^{({{\mathfrak {g}}})}_m$ at leading order in t, the statement follows immediately from (A.36) by noting that for $m\le N_{{\mathfrak {g}}}$:

$$\begin{aligned} \lim _{t\rightarrow \infty } \frac{\theta _{m-1}^{({{\mathfrak {g}}})}}{\theta _{m}^{({{\mathfrak {g}}})}}= & {} 0 \quad (m\ge 1;{{\mathfrak {g}}}=C_N^{(1)},D_{N+1}^{(2)}),\\ \lim _{t\rightarrow \infty } \frac{\theta _{m-2}^{({{\mathfrak {g}}})}}{\theta _{m}^{({{\mathfrak {g}}})}}= & {} 0 \quad (m\ge 2;{{\mathfrak {g}}}=B_N^{(1)},A_{2N-1}^{(2)},A_{2N}^{(2)}). \end{aligned}$$

For any ${{\mathfrak {g}}}$, define $\Pi _\lambda ^{({{\mathfrak {g}}})}=\lim _{t\rightarrow \infty } t^{-\rho ^*\cdot \lambda } P_\lambda ^{({{\mathfrak {g}}})}$. We refer to these as q-Whittaker functions, although they coincide with the usual definition only for the case of ${{\mathfrak {g}}}=D_N^{(1)}$ and the twisted algebras in Table 1. The spectral theorem for $D_m^{({{\mathfrak {g}}})}$ is as follows.

Theorem A.18

The functions $\Pi ^{({{\mathfrak {g}}})}_\lambda$ are common eigenfunctions of $D_m^{({{\mathfrak {g}}})}$, with

$$\begin{aligned} D_{m}^{({{\mathfrak {g}}})}(x)\, \Pi ^{({{\mathfrak {g}}})}_\lambda (x) =\Lambda ^{\omega _m^*} \, \Pi ^{({{\mathfrak {g}}})}_\lambda (x). \end{aligned}$$

Proof

The eigenvalues are the limits

$$\begin{aligned} \delta _{\lambda ;m}^{({{\mathfrak {g}}})}=\lim _{t\rightarrow \infty } (\theta _m^{({{\mathfrak {g}}})})^{-1} {{\hat{e}}}_m^{(R^*)}(s), \qquad s_i=q^{\lambda _i} t^{N+\xi _{{\mathfrak {g}}}-i}. \end{aligned}$$

The limit extracts the leading monomial of the s variables in the symmetric function, which is the term involving $s_1,s_2,...,s_m$ with positive powers. For $m=1,2,...,N_{{\mathfrak {g}}}$ this gives, using (A.30) (A.34) and (A.12):

$$\begin{aligned} \delta _{\lambda ;m}^{({{\mathfrak {g}}})}=\lim _{t\rightarrow \infty } t^{-m(N+\xi _g-\frac{m+1}{2})} {{\hat{e}}}_m(s)=q^{\lambda _1+\lambda _2+\cdots +\lambda _m}. \end{aligned}$$

By inspection, using (A.31), (A.35) and (A.32),

$$\begin{aligned}{} & {} D_N^{(1)}: \quad \delta _{\lambda ;N}^{(D_{N}^{(1)})}=q^{\frac{1}{2}(\lambda _1+\lambda _2+\cdots +\lambda _{N-1}+\lambda _N)},\ \delta _{\lambda ;N-1}^{(D_{N}^{(1)})}=q^{\frac{1}{2}(\lambda _1+\lambda _2+\cdots +\lambda _{N-1}-\lambda _N)},\\{} & {} C_N^{(1)}: \quad \delta _{\lambda ;N}^{(C_N^{(1)})}=q^{\frac{1}{2}(\lambda _1+\lambda _2+\cdots +\lambda _{N-1}+\lambda _N)},\\{} & {} D_{N+1}^{(2)}: \delta _{\lambda ;N}^{(D_{N+1}^{(2)})} =q^{\frac{1}{2}(\lambda _1+\lambda _2+\cdots +\lambda _{N-1}+\lambda _N)}. \end{aligned}$$

The result can be uniformly written as $\Lambda ^{\omega _m^*}=q^{\omega _m^*\cdot \lambda }$, $\omega _m^*$ the fundamental weights of $R^*$. $\square$

Remark A.19

Using the convention $V_0^{({{\mathfrak {g}}})}=1$, Def. A.8 implies

$$\begin{aligned} D_{m}^{({{\mathfrak {g}}})}= \sum _{j=0}^m V_j^{({{\mathfrak {g}}})}, \qquad m\le N_{{\mathfrak {g}}}. \end{aligned}$$

(A.39)

Appendix B: Pieri rules

In this section, we list some of the Pieri operators implementing the first Pieri rule for Koornwinder and ${{\mathfrak {g}}}$-Macdonald polynomials. Their q-Whittaker limit gives q-difference Toda Hamiltonians for certain root systems.

1.1 First Pieri rule for Koornwinder polynomials

The first Pieri operator for generic (a, b, c, d) is the q-difference operator ${{\mathcal {H}}}_1^{(a,b,c,d)}(s)$ of Theorem (3.3).

Theorem B.1

The first Pieri operator for Koornwinder polynomials is

$$\begin{aligned} {{\mathcal {H}}}_1^{(a,b,c,d)}(s;q,t)= & {} a^{-1}t^{1-N}\varphi ^{(a^*,b^*,c^*,d^*)}(s)+ \sum _{i=1}^N\left[ \left( \prod _{j=1}^{i-1}\frac{s_i-t^{-1}s_j }{s_i-s_j t^{i-j}}\frac{q s_i-t s_j}{q s_i-s_j}\right) T_i\right. \\{} & {} + \frac{\prod _{u\in \{a^*,b^*,c^*,d^*\}}(1-u^{-1}s_i)(q- u s_i)}{(1-s_i^2)(q-s_i^2)^2(q^2-s_i^2)} \\{} & {} \times \left. \left( \prod _{j=i+1}^{N} \frac{t^{-1}s_i-s_j}{s_i-s_j}\frac{t s_i-q s_j}{s_i-q s_j} \prod _{j\ne i} \frac{q-t s_i s_j }{q-s_i s_j}\frac{1-t^{-1} s_i s_j}{1-s_i s_j}\right) T_i^{-1} \right] \!, \end{aligned}$$

where

$$\begin{aligned} \varphi ^{(a^*,b^*,c^*,d^*)}(x)= & {} \sum _{\epsilon =\pm 1} \frac{\big (1-\frac{\epsilon }{ q^{1/2}}a^*\big ) \big (1-\frac{\epsilon }{q^{1/2}}b^*\big ) \big (1-\frac{\epsilon }{q^{1/2}}c^*\big )\big (1- \frac{\epsilon }{q^{1/2}}d^*\big )}{2(1-t)(1-q^{-1}t)}\nonumber \\{} & {} \times \left\{ \prod _{i=1}^N \frac{1-\frac{\epsilon \, t }{q^{1/2}}x_i}{1-\frac{\epsilon }{q^{1/2}}x_i}\, \frac{1-\frac{\epsilon \, t }{q^{1/2}}x_i^{-1}}{1-\frac{\epsilon }{q^{1/2}}x_i^{-1}} -t^N\right\} , \end{aligned}$$

and $(a^*,b^*,c^*,d^*)$ are the dual Koornwinder parameters (3.11).

Proof

We use the formula (3.18) with $(a,b,c,d)\rightarrow (a^*,b^*,c^*,d^*)$. For simplicity, we work with the values (a, b, c, d) and interchange them with their duals at the end. We need the following.

Lemma B.2

$$\begin{aligned}{} & {} \Delta ^{(a,b,c,d)}(x)^{-1} \Gamma _i \Delta ^{(a,b,c,d)}(x)\\{} & {} \quad =\frac{(1-\frac{1}{x_i^2})(1-\frac{1}{q x_i^2})}{(1-\frac{1}{ax_i})(1-\frac{1}{bx_i})(1-\frac{1}{cx_i})(1-\frac{1}{dx_i})}\!\prod _{j=1}^{i-1} \frac{1-\frac{q x_i}{t x_j}}{1-\frac{q x_i}{x_j}}\! \prod _{j=i+1}^N\frac{1-\frac{x_j}{x_i}}{1-\frac{x_j}{t x_i}} \prod _{j\ne i} \frac{1-\frac{1}{x_ix_j}}{1-\frac{1}{tx_ix_j}}\, \Gamma _i,\\{} & {} \Delta ^{(a,b,c,d)}(x)^{-1} \Gamma _i^{-1} \Delta ^{(a,b,c,d)}(x)\\{} & {} \quad =\frac{(1-\frac{q}{ax_i})(1-\frac{q}{bx_i})(1-\frac{q}{cx_i})(1-\frac{q}{dx_i})}{(1-\frac{q}{x_i^2})(1-\frac{q^2}{x_i^2})}\!\prod _{j=1}^{i-1}\frac{1-\frac{x_i}{ x_j}}{1-\frac{x_i}{t x_j}}\! \prod _{j=i+1}^N\frac{1-\frac{qx_j}{ tx_i}}{1-\frac{qx_j}{x_i}} \prod _{j\ne i} \frac{1-\frac{q}{t x_ix_j}}{1-\frac{q}{x_ix_j}}\, \Gamma _i^{-1}\!. \end{aligned}$$

Using Lemma A.11, the conjugation of the ${{\mathcal {D}}}_1^{(a,b,c,d)}$ operator boils down to that of the terms $\Phi _{i,\epsilon }^{(a,b,c,d)} \Gamma _i^\epsilon$. We have:

$$\begin{aligned}{} & {} \Delta ^{(a,b,c,d)}(x)^{-1} \Phi _{i,+}^{(a,b,c,d)} \Gamma _i\Delta ^{(a,b,c,d)}(x)=\frac{a b c d}{q} t^{2N-2} \left( \prod _{j=1}^{i-1} \frac{1-\frac{x_j}{tx_i}}{1-\frac{x_j}{x_i}}\frac{1-\frac{qx_i}{t x_j}}{1-\frac{qx_i}{x_j}}\right) \Gamma _i,\\{} & {} \Delta ^{(a,b,c,d)}(x)^{-1} \Phi _{i,-}^{(a,b,c,d)} \Gamma _i^{-1}\Delta ^{(a,b,c,d)}(x)\\{} & {} =t^{2N-2} \frac{(1-\frac{a}{x_i})(1-\frac{q}{ax_i})(1-\frac{b}{x_i})(1-\frac{q}{bx_i})(1-\frac{c}{x_i})(1-\frac{q}{cx_i})(1-\frac{d}{x_i})(1-\frac{q}{dx_i})}{(1-\frac{1}{x_i^2})(1-\frac{q}{x_i^2})^2(1-\frac{q^2}{x_i^2})}\\{} & {} \qquad \qquad \quad \times \, \left( \prod _{j=i+1}^{N} \frac{1-\frac{x_i}{t x_j}}{1-\frac{x_i}{ x_j}}\frac{1-\frac{qx_j}{t x_i}}{1-\frac{qx_j}{x_i}}\prod _{j\ne i} \frac{1-\frac{q}{t x_ix_j}}{1-\frac{q}{x_ix_j}}\frac{1-\frac{x_ix_j}{t}}{1-x_ix_j}\right) \Gamma _i^{-1}. \end{aligned}$$

We finally use the formula (3.18) for $m=1$ and substitute the above results with the change $(a,b,c,d)\rightarrow (a^*,b^*,c^*,d^*)$, and the Theorem follows. $\square$

1.2 First Pieri rules for ${{\mathfrak {g}}}$

We may now specialize the result of Theorem B.1 to ${{\mathfrak {g}}}$, according to Table 1. We first treat the cases where ${{\mathfrak {g}}}=D_N^{(1)},C_N^{(1)},A_{2N-1}^{(2)}$ for which the constant term $\varphi ^{({{\mathfrak {g}}}^*)}(x)$ vanishes¹⁰:

$$\begin{aligned} {{\mathcal {H}}}_1^{(D_N^{(1)})}= & {} \sum _{i=1}^N\left\{ \prod _{j<i} \frac{t^{i-j-1} {\Lambda }_j -{\Lambda }_i}{t^{i-j}{\Lambda }_j-{\Lambda }_i}\frac{t^{i-j+1} {\Lambda }_j -q {\Lambda }_i}{t^{i-j}{\Lambda }_j-q {\Lambda }_i} \, T_i \right. \\{} & {} +\!\prod _{j>i}\! \frac{t^{j-i+1} {\Lambda }_i-q{\Lambda }_j}{t^{j-i}{\Lambda }_i-q{\Lambda }_j}\frac{t^{j-i-1} {\Lambda }_i-{\Lambda }_j}{t^{j-i}{\Lambda }_i-{\Lambda }_j}\\{} & {} \left. \times \prod _{j\ne i}\!\!\frac{1-t^{2N-i-j-1}{\Lambda }_i{\Lambda }_j}{1-t^{2N-i-j}{\Lambda }_i{\Lambda }_j} \frac{t^{2N-i-j+1}{\Lambda }_i{\Lambda }_j-q}{t^{2N-i-j}{\Lambda }_i {\Lambda }_j-q} T_i^{-1}\!\!\right\} \!,\\ {{\mathcal {H}}}_1^{(C_N^{(1)})}= & {} \sum _{i=1}^N \left\{ \prod _{j<i} \frac{t^{j-i+1} {\Lambda }_i-{\Lambda }_j}{t^{j-i} {\Lambda }_i-{\Lambda }_j}\frac{{\Lambda }_j-qt^{j-i-1}{\Lambda }_i}{{\Lambda }_j-qt^{j-i}{\Lambda }_i} \,T_i \right. \\{} & {} + \frac{1 -t^{N-i}{\Lambda }_i}{1-t^{N+1-i}{\Lambda }_i}\frac{t^{N+2-i}{\Lambda }_i-q}{t^{N+1-i}{\Lambda }_i-q} \prod _{j>i}\frac{t^{j-i+1} {\Lambda }_i-q{\Lambda }_j}{t^{j-i} {\Lambda }_i-q{\Lambda }_j}\frac{{\Lambda }_j-t^{j-i-1}{\Lambda }_i}{{\Lambda }_j-t^{j-i}{\Lambda }_i}\\{} & {} \times \left. \prod _{j\ne i} \frac{1 -t^{2N+1-i-j}{\Lambda }_i {\Lambda }_j}{1-t^{2N+2-i-j}{\Lambda }_i{\Lambda }_j} \frac{t^{2N+3-i-j}{\Lambda }_i{\Lambda }_j-q}{t^{2N+2-i-j}{\Lambda }_i{\Lambda }_j-q}\, T_i^{-1} \right\} . \\ {{\mathcal {H}}}_1^{(A_{2N-1}^{(2)})}= & {} \sum _{i=1}^N\left\{ \prod _{j<i} \frac{t^{j-i+1}\Lambda _i -\Lambda _j}{t^{j-i}\Lambda _i -\Lambda _j} \frac{{\Lambda }_j-q t^{j-i-1}{\Lambda }_i}{{\Lambda }_j-q t^{j-i}{\Lambda }_i}\, T_i\right. \\{} & {} \left. +\frac{t^{2N-2i}\Lambda _i^2-1}{t^{2N+1-2i}\Lambda _i^2-1} \frac{t^{2N+2-2i}\Lambda _i^2-q^2}{t^{2N+1-2i}\Lambda _i^2-q^2} \prod _{j>i} \frac{t^{j-i+1}\Lambda _i-q \Lambda _j}{t^{j-i}\Lambda _i-q \Lambda _j}\frac{t^{j-i-1}\Lambda _i-\Lambda _j}{t^{j-i}\Lambda _i-\Lambda _j} \right. \\{} & {} \left. \times \prod _{j\ne i} \frac{t^{2N-i-j}\Lambda _i\Lambda _j-1}{t^{2N+1-i-j}\Lambda _i\Lambda _j-1} \frac{t^{2N+2-i-j}\Lambda _i\Lambda _j-q}{t^{2N+1-i-j}\Lambda _i\Lambda _j-q}\, T_i^{-1}\right\} . \end{aligned}$$

We now treat the cases ${{\mathfrak {g}}}=B_N^{(1)},D_{N+1}^{(2)},A_{2N}^{(2)}$ for which $\varphi ^{({{\mathfrak {g}}}^*)}$ has a non-trivial contribution. We find:

$$\begin{aligned} {{\mathcal {H}}}_1^{(B_N^{(1)})}= & {} G^{(B_N^{(1)})}(\Lambda )+\sum _{i=1}^N\left\{ \prod _{j< i} \frac{t^{j-i+1} {\Lambda }_i-{\Lambda }_j}{t^{j-i} {\Lambda }_i-{\Lambda }_j}\frac{{\Lambda }_j-qt^{j-i-1}{\Lambda }_i}{ {\Lambda }_j-qt^{j-i}{\Lambda }_i}\,T_i\right. \\{} & {} +\frac{1 -t^{2N-2i}{\Lambda }_i^2}{1-t^{2N+1-2i}{\Lambda }_i^2}\,\frac{q -t^{2N-2i}{\Lambda }_i^2}{q -t^{2N+1-2i}{\Lambda }_i^2}\,\frac{t^{2N+2-2i}{\Lambda }_i^2-q^2}{t^{2N+1-2i}{\Lambda }_i^2-q^2}\,\frac{t^{2N+2-2i}{\Lambda }_i^2-q}{t^{2N+1-2i}{\Lambda }_i^2-q}\\{} & {} \times \prod _{j>i} \!\!\frac{t^{j-i+1} {\Lambda }_i-q{\Lambda }_j}{t^{j-i} {\Lambda }_i-q{\Lambda }_j}\frac{{\Lambda }_j-t^{j-i-1}{\Lambda }_i}{{\Lambda }_j-t^{j-i}{\Lambda }_i}\!\\{} & {} \left. \times \prod _{j\ne i} \!\frac{1 -t^{2N-i-j}{\Lambda }_i {\Lambda }_j}{1-t^{2N+1-i-j}{\Lambda }_i {\Lambda }_j} \frac{t^{2N+2-i-j}{\Lambda }_i {\Lambda }_j-q}{t^{2N+1-i-j}{\Lambda }_i {\Lambda }_j-q} T_i^{-1} \!\right\} , \end{aligned}$$

where

$$\begin{aligned} G^{(B_N^{(1)})}({\Lambda })= & {} -1+\frac{1}{2}\sum _{\epsilon =\pm 1} \prod _{i=1}^N \frac{t^{N+\frac{3}{2}-i}\Lambda _i -\epsilon q^{1/2}}{t^{N+\frac{1}{2}-i}\Lambda _i -\epsilon q^{1/2}}\, \frac{t^{N-\frac{1}{2}-i}\Lambda _i -\epsilon q^{-1/2}}{t^{N+\frac{1}{2}-i}\Lambda _i -\epsilon q^{-1/2}}. \\ {{\mathcal {H}}}_1^{(D_{N+1}^{(2)})}= & {} G^{(D_{N+1}^{(2)})}({\Lambda })+\sum _{i=1}^N\left\{ \prod _{j<i} \frac{t^{j-i+1}{\Lambda }_i-{\Lambda }_j}{t^{j-i}{\Lambda }_i-{\Lambda }_j} \frac{qt^{j-i-1}{\Lambda }_i-{\Lambda }_j}{qt^{j-i}{\Lambda }_i-{\Lambda }_j}\, T_i\right. \\{} & {} +\frac{1-t^{N-i}\Lambda _i}{1-t^{N+1-i}\Lambda _i} \frac{q^{1/2}-t^{N-i}\Lambda _i}{q^{1/2}-t^{N+1-i}\Lambda _i} \frac{q-t^{N+2-i}\Lambda _i}{q-t^{N+1-i}\Lambda _i} \frac{q^{1/2}-t^{N+2-i}\Lambda _i}{q^{1/2}-t^{N+1-i}\Lambda _i}\\{} & {} \times \prod _{j\ne i} \!\!\frac{1-t^{2N+1-i-j}{\Lambda }_i{\Lambda }_j}{1-t^{2N+2-i-j}{\Lambda }_i{\Lambda }_j} \frac{q-t^{2N+3-i-j}{\Lambda }_i{\Lambda }_j}{q-t^{2N+2-i-j}{\Lambda }_i{\Lambda }_j}\\{} & {} \left. \times \prod _{j>i}\!\frac{ t^{j-i-1}{\Lambda }_i-{\Lambda }_j}{ t^{j-i}{\Lambda }_i-{\Lambda }_j} \frac{t^{j-i+1}{\Lambda }_i-q{\Lambda }_j}{t^{j-i}{\Lambda }_i-q{\Lambda }_j} T_i^{-1}\!\right\} \!, \end{aligned}$$

where

$$\begin{aligned} G^{(D_{N+1}^{(2)})}\!({\Lambda })\!=\!\frac{(1-t q^{-1/2})(1+q^{-1/2})}{1-t q^{-1}}\!\left\{ \! \prod _{i=1}^N \!\frac{q^{1/2}- {\Lambda }_it^{N+2-i}}{q^{1/2}- {\Lambda }_it^{N+1-i}}\frac{1-q^{1/2}{\Lambda }_it^{N-i}}{1-q^{1/2}{\Lambda }_it^{N+1-i}}\!-\!1\!\right\} \!, \end{aligned}$$

and finally

$$\begin{aligned} {{\mathcal {H}}}_1^{(A_{2N}^{(2)})}= & {} G^{(A_{2N}^{(2)})}({\Lambda })+\sum _{i=1}^N\left\{ \prod _{j<i} \frac{t^{j-i+1}{\Lambda }_i-{\Lambda }_j}{t^{j-i}{\Lambda }_i-{\Lambda }_j} \frac{qt^{j-i-1}{\Lambda }_i-{\Lambda }_j}{qt^{j-i}{\Lambda }_i-{\Lambda }_j}\, T_i\right. \\{} & {} +\frac{1-t^{N-i}\Lambda _i}{1-t^{N+1-i}\Lambda _i} \frac{q-t^{2N-2i+1}\Lambda _i^2}{q-t^{2N+2-2i}\Lambda _i^2} \frac{q-t^{2N+3-2i}\Lambda _i^2}{q-t^{2N+2-2i}\Lambda _i^2} \frac{q-t^{N+2-i}\Lambda _i}{q-t^{N+1-i}\Lambda _i}\\{} & {} \times \! \prod _{j>i}\!\frac{ t^{j-i-1}{\Lambda }_i\!-\!{\Lambda }_j}{ t^{j-i}{\Lambda }_i\!-\!{\Lambda }_j} \frac{t^{j-i+1}{\Lambda }_i\!-\!q{\Lambda }_j}{t^{j-i}{\Lambda }_i\!-\!q{\Lambda }_j}\\{} & {} \left. \times \prod _{j\ne i} \!\frac{1\!-\!t^{2N+1-i-j}{\Lambda }_i{\Lambda }_j}{1\!-\!t^{2N+2-i-j}{\Lambda }_i{\Lambda }_j} \frac{q\!-\!t^{2N+3-i-j}{\Lambda }_i{\Lambda }_j}{q\!-\!t^{2N+2-i-j}{\Lambda }_i{\Lambda }_j} T_i^{-1}\!\!\right\} \!, \end{aligned}$$

where

$$\begin{aligned} G^{(\!A_{2N}^{(2)})}\!(\!{\Lambda }\!)= & {} \sum _{\epsilon =\pm 1}\!\!\! \frac{(1\!-\!\epsilon t q^{-1/2})(1\!+\!\epsilon q^{-1/2})}{2(1\!-\!t q^{-1})} \!\!\left\{ \! \prod _{i=1}^N \frac{q^{1/2}\!-\!\epsilon {\Lambda }_it^{N+2-i}}{q^{1/2}\!-\!\epsilon {\Lambda }_it^{N+1-i}} \frac{\epsilon \!-\!q^{1/2}{\Lambda }_it^{N-i}}{\epsilon \!-\!q^{1/2}{\Lambda }_it^{N+1-i}}\!-\!1\!\!\right\} \!. \end{aligned}$$

Remark B.3

When we consider the q-Whittaker limit $t\rightarrow \infty$, the constant pieces $G^{({{\mathfrak {g}}})}({\Lambda })$ in the latter three cases ${{\mathfrak {g}}}=B_N^{(1)},D_{N+1}^{(2)},A_{2N}^{(2)}$ provide an extra constant term in the limiting Hamiltonians $H_1^{({{\mathfrak {g}}})}$. These read respectively:

$$\begin{aligned}{} & {} G^{(B_N^{(1)})}\!(\Lambda )\rightarrow \frac{q^{-1}}{{\Lambda }_{N-1}{\Lambda }_N}-\frac{1+q^{-1}}{{\Lambda }_N^2} ,G^{(D_{N+1}^{(2)})}\!(\Lambda )\rightarrow -\frac{1+q^{-1/2}}{{\Lambda }_N},\\{} & {} G^{(A_{2N}^{(2)})}\!(\Lambda )\rightarrow -\frac{1}{{\Lambda }_N} . \end{aligned}$$

Appendix C: Proof of Theorem 4.10

Here, we provide the proof of Theorem 4.10, that the time translation operators $g^{({{\mathfrak {g}}})}$ of Eq. (4.12) commute with the limiting first Pieri operator/Toda Hamiltonian $H_1^{({{\mathfrak {g}}})}$ of Eq. (3.38):

$$\begin{aligned} g^{({{\mathfrak {g}}})} H_1^{({{\mathfrak {g}}})}= H_1^{({{\mathfrak {g}}})}g^{({{\mathfrak {g}}})} . \end{aligned}$$

(C.1)

We divide each Hamiltonian into bulk and boundary pieces, study how g commutes with each, and then sum up the contributions to prove the commutation (C.1).

Lemma C.1

For all ${{\mathfrak {g}}}$,

$$\begin{aligned} g_T\, \left( 1-{\Lambda }^{-{\alpha }_a}\right)= & {} \left( 1-q {\Lambda }^{-{\alpha }_a}\right) T^{-{\alpha }_a} g_T \quad (a=0,1,...,N-1), \nonumber \\ \end{aligned}$$

(C.2)

$$\begin{aligned} g_{\Lambda }\, \left( 1-{\Lambda }^{-{\alpha }_a}\right) T_a= & {} \left( 1-q^{-1}{\Lambda }^{-{\alpha }_{a+1}}\right) T_{a+1}\, g_{\Lambda }\quad (a=1,2,...,N-2), \nonumber \\ \end{aligned}$$

(C.3)

$$\begin{aligned} \quad g_{\Lambda }\, \left( 1-{\Lambda }^{-{\alpha }_a}\right) T_a^{-1}= & {} \left( 1-q^{-1}{\Lambda }^{-{\alpha }_{a-1}}\right) T_a^{-1}\, g_{\Lambda }\quad (a=1,2,...,N-1) , \nonumber \\ \end{aligned}$$

(C.4)

with the convention ${\alpha }_0=0$.

1.1 Bulk and boundary terms

Each of the first Pieri operators (3.38) is the sum of a bulk and boundary piece, where the bulk piece has the same form for all ${{\mathfrak {g}}}$:

$$\begin{aligned} H_1^{[m]}:=\sum _{a=1}^m \left( 1-\frac{{\Lambda }_a}{{\Lambda }_{a-1}}\right) \, T_a+\left( 1-\frac{{\Lambda }_{a+1}}{{\Lambda }_{a}}\right) \, \frac{1}{T_a}, \end{aligned}$$

with the convention that $\frac{1}{{\Lambda }_0}=0$, for suitable value of $m= m_{{\mathfrak {g}}}$: $m_{{\mathfrak {g}}}=N-2$ for ${{\mathfrak {g}}}=D_N^{(1)},B_N^{(1)},A_{2N}^{(2)}$, and $m_{{\mathfrak {g}}}=N-1$ for all other cases.

The operators $g^{({{\mathfrak {g}}})}$ of (4.12) also have some common structures. For ${{\mathfrak {g}}}=B_N^{(1)},A_{2N-1}^{(2)},D_{N+1}^{(2)}$, $g=g_T^{\frac{1}{2}} {\alpha }g_T^{\frac{1}{2}} g_{\Lambda }\beta$ where ${\alpha },\beta$ are functions of ${\Lambda }_N$ only. For ${{\mathfrak {g}}}=D_N^{(1)}$ $g=g_T g_{\Lambda }\beta$ with $\beta$ a function of ${\Lambda }_{N-1}{\Lambda }_N$ only. The cases ${{\mathfrak {g}}}=C_N^{(1)},A_{2N}^{(2)}$ are different, and have a g operator of the form $g=g_Tg_{\Lambda }{\alpha }g_Tg_{\Lambda }\beta$, where ${\alpha },\beta$ are functions of ${\Lambda }_N$ only.

Lemma C.2

For all ${{\mathfrak {g}}}$ and $m\le m_{{\mathfrak {g}}}$, and $t_1=2$ for ${{\mathfrak {g}}}=C_N^{(1)},A_{2N}^{(2)}$, $t_1=1$ otherwise,

$$\begin{aligned} g^{({{\mathfrak {g}}})}\, H_1^{[m]} \, (g^{({{\mathfrak {g}}})})^{-1} =(g_T\,g_{\Lambda })^{t_1} \, H_1^{[m]} \, (g_T\,g_{\Lambda })^{-t_1} . \end{aligned}$$

Proof

The only subtleties arise in the cases ${{\mathfrak {g}}}=B_N^{(1)},A_{2N}^{(2)}$. When ${{\mathfrak {g}}}=B_N^{(1)}$, with $g=g_T^{\frac{1}{2}} {\alpha }g_T^{\frac{1}{2}} g_{\Lambda }\beta$, the action of $g_T^{\frac{1}{2}} g_{\Lambda }\beta$ on $H_1^{[m]}$ creates a term proportional to $T_{m+1}$, as a consequence of

$$\begin{aligned} g_T\, g_{\Lambda }\, \left( 1-\frac{{\Lambda }_a}{{\Lambda }_{a-1}}\right) \, T_a =\left( T_a-\frac{{\Lambda }_{a+1}}{{\Lambda }_a}\,T_{a+1}\right) \, g_Tg_{\Lambda }\end{aligned}$$

(C.5)

for $a=m$. This term commutes with ${\alpha }$ only if $m+1\le N-1$, hence we set $m_{{\mathfrak {g}}}=N-2$. Similarly, when ${{\mathfrak {g}}}=A_{2N}^{(2)}$, with $g=g_Tg_{\Lambda }{\alpha }g_Tg_{\Lambda }\beta$, the action of $g_Tg_{\Lambda }\beta$ on $H_1^{[m]}$ creates a term proportional to $T_{m+1}$ which commutes with ${\alpha }$ only if $m+1\le N-1$, hence we set $m_{{\mathfrak {g}}}=N-2$ as well. $\square$

Lemma C.3

We have the commutation relations for $m=1,2,...,N-1$:

$$\begin{aligned} g_T\,g_{\Lambda }\, H_1^{[m]} =\left\{ H_1^{[m]} +\frac{{\Lambda }_{m+1}}{{\Lambda }_m}\,\left( \frac{1}{T_m}- T_{m+1}\right) \right\} \, g_T \,g_{\Lambda }. \end{aligned}$$

Proof

We simply sum over the relevant values of a the commutation relations (C.5), as well as the following:

$$\begin{aligned} g_T\, g_{\Lambda }\, \left( 1-\frac{{\Lambda }_{a+1}}{{\Lambda }_{a}}\right) \, \frac{1}{T_a} =\left( \frac{1}{T_a}-\frac{{\Lambda }_{a}}{{\Lambda }_{a-1}}\, \frac{1}{T_{a-1}}\right) \, g_Tg_{\Lambda }, \end{aligned}$$

(C.6)

obtained by combining (C.2-C.4). $\square$

The cases ${{\mathfrak {g}}}=C_N^{(1)}$ and $A_{2N}^{(1)}$ are special, as we will have to use Lemma C.3twice. This is deferred to Sect. C.6 below.

For all the other cases, we apply Lemmas C.2 and C.3 to commute g through the relevant bulk contributions to $H_1^{({{\mathfrak {g}}})}$ corresponding to $a=1,2,..,m_{{\mathfrak {g}}}$. The remaining terms of $H_1^{({{\mathfrak {g}}})}$ are the “boundary” terms, corresponding to $m_{{\mathfrak {g}}}<a\le N$ and possibly to constant terms independent of $T_a^{\pm 1}$. We address them individually in the following Sections C.2 through C.5. In all cases, the Lemma 4.10 follows from summing all contributions from the bulk $a=1,2,...,m_{{\mathfrak {g}}}$ and the boundary $a=m_{{\mathfrak {g}}}+1,...,N$ plus constant terms.

1.2 Case ${{\mathfrak {g}}}=D_{N}^{(1)}$

Recall that $g=g_Tg_{\Lambda }\prod _{n=0}^\infty \left( 1-\frac{q^n}{{\Lambda }_{N-1}{\Lambda }_N}\right) ^{-1}$, and that $m_{{\mathfrak {g}}}=N-2$. For $a=N-1$, we have:

$$\begin{aligned}{} & {} g\left( 1-\frac{{\Lambda }_{N-1}}{{\Lambda }_{N-2}}\right) T_{N-1}= \left( T_{N-1} -\frac{\Lambda _{N}}{\Lambda _{N-1}}\, T_{N}-\frac{1}{\Lambda _{N-1}\Lambda _{N}} T_{N}^{-1}+\frac{1}{\Lambda _{N-1}^2} T_{N-1}^{-1}\right) \, g, \\{} & {} g\, \left( 1-\frac{\Lambda _{N}}{\Lambda _{N-1}}\right) \left( 1-\frac{1}{\Lambda _{N-1}\Lambda _{N}}\right) T_{N-1}^{-1}= \left( T_{N-1}^{-1}-\frac{\Lambda _{N-1}}{\Lambda _{N-2}} \,T_{N-2}^{-1}\right) \, g, \end{aligned}$$

and for $a=N$:

$$\begin{aligned}{} & {} g\, \left( 1-\frac{\Lambda _{N}}{\Lambda _{N-1}}\right) T_{N}=\left( T_{N} -\frac{1}{\Lambda _{N-1}\Lambda _{N}} T_{N-1}^{-1}\right) \, g,\\{} & {} g\,\left( 1-\frac{1}{\Lambda _{N-1}\Lambda _{N}}\right) \, T_N^{-1}=\left( T_{N}^{-1}-\frac{\Lambda _{N}}{\Lambda _{N-1}}\, T_{N-1}^{-1}\right) \, g. \end{aligned}$$

Eq. (C.1) follows by summing all bulk and boundary contributions.

1.3 Case ${{\mathfrak {g}}}=B_{N}^{(1)}$

Recall that $g=g_1\,g_2$, $g_1=g_T^{\frac{1}{2}} {\alpha }$, $g_2= g_1 g_{\Lambda }$ with ${\alpha }= \prod _{n=0}^\infty \left( 1-\frac{q^n}{{\Lambda }_N^2}\right) ^{-1}$, and that $m_{{\mathfrak {g}}}=N-2$. The terms $a=N-1,N$ and the constant term read

$$\begin{aligned} g \left( \!1\!-\!\frac{\Lambda _{N}}{\Lambda _{N-1}}\!\right) \!T_{N-1}^{-1}= & {} T_{N-1}^{-1}\, g_T^{\frac{1}{2}} \,{\alpha }g_T^{\frac{1}{2}} \, \left( 1-\frac{\Lambda _{N-1}}{\Lambda _{N-2}}\right) {\alpha }\,g_{\Lambda }\nonumber \\= & {} \left( T_{N-1}^{-1}-\frac{\Lambda _{N-1}}{\Lambda _{N-2}}T_{N-2}^{-1}\right) \!g,\nonumber \\ g \left( \!1\!-\!\frac{\Lambda _{N-1}}{\Lambda _{N-2}}\!\right) \!T_{N-1}= & {} T_{N-1}g_1 g_T^{\frac{1}{2}}\!\! \left( \!1\!-\!\frac{\Lambda _{N}}{\Lambda _{N-1}}\!\right) \!{\alpha }g_{\Lambda } \!\nonumber \\= & {} g_1\!\!\left( \!T_{N-1}\!-\!q^{-\frac{1}{2}}\!\frac{\Lambda _{N}}{\Lambda _{N-1}}T_{N-1}^\frac{1}{2}T_N^{\frac{1}{2}}\!\right) \! g_2,\nonumber \\ g \left( 1-\frac{\Lambda _{N}}{\Lambda _{N-1}}\right) \,T_N= & {} g_{1}\, \left( 1-\frac{1}{\Lambda _{N}^2}T_N^{-1}\right) \left( 1-q^{-1}\frac{1}{\Lambda _{N}^2}T_N^{-1}\right) \, T_N\,g_{2},\nonumber \\ g\left( 1-\frac{1}{\Lambda _{N}^2}\right) \left( 1-q\frac{1}{\Lambda _{N}^2}\right) \,T_N^{-1}= & {} g_1\, \left( T_N^{-1}-q^{-\frac{1}{2}}\frac{\Lambda _{N}}{\Lambda _{N-1}}T_{N-1}^{-\frac{1}{2}}T_{N}^{-\frac{1}{2}}\right) \, g,\nonumber \\ g \left( \frac{q^{-1}}{\Lambda _{N-1}\Lambda _N}-\frac{1+q^{-1}}{\Lambda _{N}^2}\right)= & {} g_{1} \left( \frac{q^{-\frac{1}{2}}}{\Lambda _{N-1}\Lambda _N}T_{N-1}^{-\frac{1}{2}}T_{N}^{-\frac{1}{2}}-q\frac{1+q^{-1}}{\Lambda _{N}^2}T_N^{-1}\right) \,g_{2} . \nonumber \\ \end{aligned}$$

(C.7)

Summing the last four terms leads to:

$$\begin{aligned}{} & {} g\left\{ \left( 1-\frac{\Lambda _{N-1}}{\Lambda _{N-2}}\right) \,T_{N-1}+ \left( 1-\frac{\Lambda _{N}}{\Lambda _{N-1}}\right) \,T_N \right. \\{} & {} \qquad +\left. \left( 1-\frac{1}{\Lambda _{N}^2}\right) \left( 1-q\frac{1}{\Lambda _{N}^2}\right) \,T_N^{-1}+ \frac{q^{-1}}{\Lambda _{N-1}\Lambda _N}-\frac{1+q^{-1}}{\Lambda _{N}^2} \right\} \\{} & {} \quad = g_1\left\{ \left( 1-\frac{1}{\Lambda _{N}^2}\right) \left( 1-q\frac{1}{\Lambda _{N}^2}\right) \,T_N^{-1} -\left( 1-\frac{1}{\Lambda _N^2}\right) q^{-\frac{1}{2}}\frac{\Lambda _N}{\Lambda _{N-1}} T_{N-1}^{-\frac{1}{2}}T_{N}^{-\frac{1}{2}}\right. \\{} & {} \qquad + \left. T_{N-1}-q^{-\frac{1}{2}}\frac{\Lambda _{N}}{\Lambda _{N-1}}T_N^{\frac{1}{2}} T_{N-1}^{\frac{1}{2}}+T_N-\frac{1+q^{-1}}{\Lambda _{N}^2} \right\} g_2\\{} & {} \quad =\left\{ T_{N-1}+\left( 1-\frac{\Lambda _N}{\Lambda _{N-1}}\right) T_N-\frac{\Lambda _N}{\Lambda _{N-1}}T_{N-1}^{-1}\right. \\{} & {} \qquad +\left. \left( 1-\frac{1}{\Lambda _{N}^2}\right) \left( 1-q\frac{1}{\Lambda _{N}^2}\right) \,T_N^{-1}+\frac{q^{-1}}{\Lambda _{N-1}\Lambda _N}-\frac{1+q^{-1}}{\Lambda _N^2} \right\} \, g . \end{aligned}$$

Finally adding the first term of (C.7) gives

$$\begin{aligned}{} & {} g\left\{ \left( 1-\frac{\Lambda _{N}}{\Lambda _{N-1}}\right) \,T_{N-1}^{-1}+\left( 1-\frac{\Lambda _{N-1}}{\Lambda _{N-2}}\right) \,T_{N-1}+ \left( 1-\frac{\Lambda _{N}}{\Lambda _{N-1}}\right) \,T_N \right. \\{} & {} \qquad +\left. \left( 1-\frac{1}{\Lambda _{N}^2}\right) \left( 1-q\frac{1}{\Lambda _{N}^2}\right) \,T_N^{-1}+ \frac{q^{-1}}{\Lambda _{N-1}\Lambda _N}-\frac{1+q^{-1}}{\Lambda _{N}^2} \right\} \\{} & {} \quad =\left\{ T_{N-1}+\left( 1-\frac{\Lambda _N}{\Lambda _{N-1}}\right) T_N-\frac{{\Lambda }_{N-1}}{{\Lambda }_{N-2}}T_{N-2}^{-1}+\left( 1-\frac{\Lambda _N}{\Lambda _{N-1}}\right) T_{N-1}^{-1}\right. \\{} & {} \qquad +\left. \left( 1-\frac{1}{\Lambda _{N}^2}\right) \left( 1-q\frac{1}{\Lambda _{N}^2}\right) \,T_N^{-1}+\frac{q^{-1}}{\Lambda _{N-1}\Lambda _N}-\frac{1+q^{-1}}{\Lambda _N^2} \right\} \, g. \end{aligned}$$

Eq. (C.1) follows by summing all bulk and boundary contributions.

1.4 Case ${{\mathfrak {g}}}=A_{2N-1}^{(2)}$

Recall that $g=g_Tg_{\Lambda }{\alpha }$, with ${\alpha }=\prod _{n=0}^\infty \left( 1-\frac{q^{2n}}{{\Lambda }_N^2}\right) ^{-1}$, and $m_{{\mathfrak {g}}}=N-1$. For the two boundary terms for $a=N$ we have:

$$\begin{aligned} g\,\left( 1-\frac{\Lambda _N}{\Lambda _{N-1}}\right) T_N= & {} g_T\, \left( 1-q^{-2}\frac{1}{\Lambda _N^2}\right) T_N\, g_{\Lambda }\,{\alpha }=\left( T_N-\frac{1}{\Lambda _N^2}T_{N}^{-1}\right) \,g,\\ g\,\left( 1-\frac{1}{\Lambda _{N}^2}\right) T_N^{-1}\, g^{-1}= & {} g_T\,\left( 1-q^{-1}\frac{\Lambda _{N}}{\Lambda _{N-1}}\right) T_N^{-1} \,g_{\Lambda }\,{\alpha }= \left( T_N^{-1}-\frac{\Lambda _{N}}{\Lambda _{N-1}}T_{N-1}^{-1}\right) \, g. \end{aligned}$$

Eq. (C.1) follows by summing all bulk and boundary contributions.

1.5 Case ${{\mathfrak {g}}}=D_{N+1}^{(2)}$

Recall that $g=g_Tg_{\Lambda }{\alpha }$, with ${\alpha }=\prod _{n=0}^\infty \left( 1-\frac{q^{\frac{n}{2}}}{{\Lambda }_N}\right) ^{-1}$, and $m_{{\mathfrak {g}}}=N-1$. For the two boundary terms for $a=N$ and the constant term we have:

$$\begin{aligned} g\,\left( 1-\frac{\Lambda _N}{\Lambda _{N-1}}\right) T_N= & {} \left( T_N-\frac{1+q^{-\frac{1}{2}}}{\Lambda _N}+\frac{q}{\Lambda _N^{2}}T_N^{-1}\right) \, g\\ g\,\left( 1-\frac{1}{\Lambda _{N}}\right) \left( 1-\frac{q^\frac{1}{2}}{\Lambda _{N}}\right) T_N^{-1}= & {} \left( T_N^{-1}-\frac{\Lambda _{N}}{\Lambda _{N-1}}T_{N-1}^{-1}\right) \, g\\ g\,\left( -\frac{1+q^{-\frac{1}{2}}}{\Lambda _{N}}\right)= & {} -\frac{1+q^\frac{1}{2}}{\Lambda _{N}}T_N^{-1}\, g \end{aligned}$$

Eq. (C.1) follows by summing all bulk and boundary contributions.

1.6 Cases ${{\mathfrak {g}}}=C_{N}^{(1)},A_{2N}^{(2)}$

Recall the general structure of the operator $g=g_1 g_2$ with $g_1=g_T g_{\Lambda }{\alpha }$ and $g_2=g_T g_{\Lambda }\beta$, where ${\alpha },\beta$ are functions of ${\Lambda }_N$ only. We note that $g_2$ is the same for both cases, with $\beta =\prod _{n=0}^\infty \left( 1-\frac{q^n}{{\Lambda }_N}\right) ^{-1}$. We note also that $H_1^{[N-1]}$ is the same for both cases, and Lemma C.3 gives the commutation of $g_2$ through $H_1^{[N-1]}$:

$$\begin{aligned} g_2\, H_1^{[N-1]}= \left\{ H_1^{[N-1]} +\frac{{\Lambda }_{N}}{{\Lambda }_{N-1}}\,\left( \frac{1}{T_{N-1}}- T_{N}\right) \right\} \, g_2. \end{aligned}$$

(C.8)

We now need to add the boundary and constant terms. For ${{\mathfrak {g}}}=C_N^{(1)}$ we have for $a=N$ (and no constant term):

$$\begin{aligned} g_{2}\, \left( 1-\frac{\Lambda _{N}}{\Lambda _{N-1}}\right) T_N= & {} \left( T_N-\frac{q^{-\frac{1}{2}}}{\Lambda _{N}}\right) \, g_{2}\\ g_{2}\, \left( 1-\frac{1}{\Lambda _{N}}\right) T_N^{-1}= & {} \left( T_N^{-1}-\frac{\Lambda _{N}}{\Lambda _{N-1}}T_{N-1}^{-1}\right) \, g_{2} . \end{aligned}$$

For ${{\mathfrak {g}}}=A_{2N}^{(2)}$ we have for $a=N$ and the constant term:

$$\begin{aligned} g_2\, \left( 1- \frac{{\Lambda }_N}{{\Lambda }_{N-1}}\right) T_N= & {} g_T\,\left( 1- \frac{q^{-1}}{{\Lambda }_N}\right) T_N\,g_{\Lambda }\,\beta =\left( T_N-\frac{q^{-\frac{1}{2}}}{{\Lambda }_N}\right) \, g_2,\\ g_2\, \left( 1- \frac{1}{{\Lambda }_N}\right) T_N^{-1}= & {} \left( T_N^{-1}-\frac{{\Lambda }_N}{{\Lambda }_{N-1}}T_{N-1}^{-1}\right) \, g_2,\\ g_2\, \left( 1- \frac{1}{{\Lambda }_N}\right)= & {} \left( 1- \frac{q^\frac{1}{2}}{{\Lambda }_N}T_N^{-1}\right) \, g_2. \end{aligned}$$

Adding these to the bulk contributions (C.8), we deduce the following commutations:

$$\begin{aligned} {{\mathfrak {g}}}=C_N^{(1)}:{} & {} g_2 H_1=\left\{ H_1+ \frac{1}{{\Lambda }_N}(T_N^{-1}-q^{-\frac{1}{2}})\right\} g_2, \end{aligned}$$

(C.9)

$$\begin{aligned} {{\mathfrak {g}}}=A_{2N}^{(2)}:{} & {} g_2 H_1=\left\{ H_1+ \frac{1-q^\frac{1}{2}}{{\Lambda }_N}(T_N^{-1}-q^{-\frac{1}{2}})\right\} g_2. \end{aligned}$$

(C.10)

We must now commute $g_1$ through this. For both cases, we split again $H_1$ into a bulk piece $H_1^{[N-1]}$ and boundary pieces and constant terms. For the bulk contribution, we use again Lemma C.3 to write in both cases

$$\begin{aligned} g_1\, H_1^{[N-1]}= \left\{ H_1^{[N-1]} +\frac{{\Lambda }_{N}}{{\Lambda }_{N-1}}\,\left( \frac{1}{T_{N-1}}- T_{N}\right) \right\} \, g_1. \end{aligned}$$

(C.11)

Then for ${{\mathfrak {g}}}=C_N^{(1)}$ and $a=N$:

$$\begin{aligned} g_{1}\, \left( 1-\frac{\Lambda _{N}}{\Lambda _{N-1}}\right) T_N= & {} T_N\, g_{1},\\ g_{1}\, \left( 1-\frac{1}{\Lambda _{N}}\right) T_N^{-1}= & {} \left( 1-\frac{q^\frac{1}{2}}{\Lambda _{N}}T_N^{-1}\right) \left( T_N^{-1}-\frac{\Lambda _{N}}{\Lambda _{N-1}}T_{N-1}^{-1}\right) \, g_{1}, \end{aligned}$$

and for ${{\mathfrak {g}}}=A_{2N}^{(2)}$, $a=N$ and the constant term:

$$\begin{aligned} g_1\, \left( 1- \frac{{\Lambda }_N}{{\Lambda }_{N-1}}\right) T_N= & {} g_T\,\left( 1- \frac{q^{-\frac{1}{2}}}{{\Lambda }_N}\right) T_N\, g_{\Lambda }\,{\alpha }=\left( T_N-\frac{1}{{\Lambda }_N}\right) \, g_1,\\ g_1\, \left( 1- \frac{q^\frac{1}{2}}{{\Lambda }_N}\right) T_N^{-1}= & {} \left( T_N^{-1}-\frac{{\Lambda }_N}{{\Lambda }_{N-1}}T_{N-1}^{-1}\right) \, g_1,\\ g_1\, \left( 1- \frac{q^{-\frac{1}{2}}}{{\Lambda }_N}\right)= & {} \left( 1- \frac{1}{{\Lambda }_N}T_N^{-1}\right) \, g_1. \end{aligned}$$

Summing these with the bulk contributions (C.9)and (C.10) respectively, we arrive at

$$\begin{aligned} {{\mathfrak {g}}}=C_N^{(1)}:{} & {} g_1 \,\left\{ H_1+ \frac{1}{{\Lambda }_N}(T_N^{-1}-q^{-\frac{1}{2}})\right\} =H_1\, g_1, \end{aligned}$$

(C.12)

$$\begin{aligned} {{\mathfrak {g}}}=A_{2N}^{(2)}:{} & {} g_1\,\left\{ H_1+ \frac{1-q^{\frac{1}{2}}}{{\Lambda }_N}(T_N^{-1}-q^{-\frac{1}{2}})\right\} =H_1\, g_1. \end{aligned}$$

(C.13)

Finally, Lemma 4.10 follows for ${{\mathfrak {g}}}=C_N^{(1)},A_{2N}^{(2)}$ by combining the commutation relations (C.9-C.10) and (C.12-C.13). In all cases, Eq. (C.1) follows by summing all bulk and boundary contributions.