(4.11-4.12) for some initial values of $n$. Using the form of
$\bar{\mathcal{T}}_{a;n}$, we see that for long roots, it is enough to show
that (4.11) holds for $n=-1$ and $n=0$, so we need the explicit expressions
for $\bar{D}_{a,n}$ with $n=-1,0,1$ to start the induction, and for short
roots, we also need the expression for $n=2$:

(4.15) $\displaystyle\bar{D}_{a,-1}$ $\displaystyle=$
$\displaystyle\bar{D}_{a,0}(1-\Lambda^{-\alpha_{a}^{*}})T^{-\omega_{a}},\qquad
a\in[1,N],$ (4.16) $\displaystyle\bar{D}_{a,2}$ $\displaystyle=$
$\displaystyle\bar{D}_{a,1}(1-\Lambda^{-\alpha_{a}^{*}}T^{-\alpha_{a}})T^{\omega_{a}},\qquad\hbox{$a$
short.}$

Let $h_{{\alpha}}(u)=({\Lambda}^{-{\alpha}};q^{u})_{\infty}^{-1}$,
$h_{\alpha}=h_{\alpha}(1)$ and
$h_{{\alpha}}^{+}=(q^{\frac{1}{2}}{\Lambda}^{-{\alpha}};q)_{\infty}^{-1}$.
Then $g_{\Lambda}=\prod_{a=1}^{N-1}h_{{\alpha}_{a}^{*}}$. It is useful to
rewrite the $g$ operators of (4.13) as

$\displaystyle g$ $\displaystyle=$ $\displaystyle
g_{T}\,g_{\Lambda}\,h_{{\alpha}_{N}^{*}}(u),\quad{\mathfrak{g}}=D_{N}^{(1)},A_{2N-1}^{(2)},D_{N+1}^{(2)},\
u=1,2,\frac{1}{2}\ {\rm resp.},$ $\displaystyle g$ $\displaystyle=$
$\displaystyle g_{1}g_{2}\ \hbox{ where }\
g_{1}=\left\\{\begin{array}[]{l}g_{T}^{\frac{1}{2}}\,h_{{\alpha}_{N}^{*}},\\\
g_{T}\,g_{\Lambda},\\\ g_{T}\,g_{\Lambda}\,h_{{\alpha}_{N}^{*}}^{+},\\\
\end{array}\right.\ \hbox{ and }\
g_{2}=\left\\{\begin{array}[]{ll}g_{1}\,g_{\Lambda},&{\mathfrak{g}}=B_{N}^{(1)},\\\
g_{1}\,h_{{\alpha}_{N}^{*}},&{\mathfrak{g}}=C_{N}^{(1)},\\\
g_{T}g_{\Lambda}h_{{\alpha}_{N}^{*}},&{\mathfrak{g}}=A_{2N}^{(2)},\end{array}\right.$

We use the exchange relations, valid for any $x,y\in\mathbb{Q}$,

(4.18) $\displaystyle(g_{T})^{x}\,({\Lambda}_{a})^{y}$ $\displaystyle=$
$\displaystyle
q^{\frac{xy^{2}}{2}}\,({\Lambda}_{a})^{y}\,(T_{a})^{xy}\,(g_{T})^{x},$ (4.19)
$\displaystyle h_{{\alpha}_{b}^{*}}\,T^{\omega_{a}}$ $\displaystyle=$
$\displaystyle
T^{\omega_{a}}\,(1-{\Lambda}^{-{\alpha}_{a}^{*}})\,h_{{\alpha}_{b}^{*}},\qquad
h_{{\alpha}_{b}^{*}}^{-1}\,T^{-\omega_{a}}=(1-{\Lambda}^{-{\alpha}_{a}^{*}})T^{-\omega_{a}}\,h_{{\alpha}_{b}^{*}}^{-1}.$

Long labels. In this case, we need only verify that (4.11) hold for $n=\pm 1$,
Note that $\bar{D}_{a;0}=\Lambda^{\omega_{a}^{*}}$ commutes with
$g_{\Lambda}$. In the self-dual cases
${\mathfrak{g}}=D_{N}^{(1)},A_{2N-1}^{(2)},D_{N+1}^{(2)}$, all labels are long
and Equation (4.11) with $n=1$,
$g\bar{D}_{a;0}g^{-1}=q^{\Lambda_{a,a}/2}\bar{D}_{a;1}$, follows from

$g_{T}\,\bar{D}_{a;0}=q^{\frac{\Lambda_{a,a}}{2}}{\Lambda}^{\omega_{a}^{*}}T^{\omega_{a}}g_{T}=q^{\frac{\Lambda_{a,a}}{2}}\bar{D}_{a;1}\,g_{T}.$

For $n=-1$ it is equivalent to

$\displaystyle g^{-1}\,\bar{D}_{a;0}$ $\displaystyle=$ $\displaystyle
h_{{\alpha}_{N}^{*}}^{-1}g_{\Lambda}^{-1}g_{T}^{-1}\,{\Lambda}^{\omega_{a}^{*}}=q^{-\frac{{\Lambda}_{a,a}}{2}}\,{\Lambda}^{\omega_{a}^{*}}\,g_{\Lambda}^{-1}T^{-\omega_{a}}\,h_{{\alpha}_{N}^{*}}^{-1}\,g_{\Lambda}^{-1}$
$\displaystyle=$ $\displaystyle
q^{-\frac{{\Lambda}_{a,a}}{2}}\,\bar{D}_{a;0}\,(1-{\Lambda}^{-{\alpha}_{a}^{*}})T^{-\omega_{a}}\,g^{-1}=q^{-\frac{{\Lambda}_{a,a}}{2}}\,\bar{D}_{a;-1}\,g^{-1}.$

For the long labels $a<N$ of $B_{N}^{(1)}$, $\omega_{a}^{*}=\omega_{a}$,
$\Lambda_{a,a}=a$ and (4.11) with $n=\pm 1$ follow from

$g_{T}^{1/2}\,h_{{\alpha}_{N}^{*}}\,g_{T}^{1/2}\,\bar{D}_{a;0}=q^{\frac{a}{4}}g_{T}^{1/2}\,{\Lambda}^{\omega_{a}^{*}}\,T^{\omega_{a}/2}\,h_{{\alpha}_{N}^{*}}\,g_{T}^{1/2}\\\
=q^{\frac{a}{2}}\,{\Lambda}^{\omega_{a}^{*}}T^{\omega_{a}}g_{T}^{1/2}\,h_{{\alpha}_{N}^{*}}\,g_{T}^{1/2}=q^{\frac{a}{2}}\bar{D}_{a;1}g_{T}^{1/2}\,h_{{\alpha}_{N}^{*}}\,g_{T}^{1/2}$

and

$\displaystyle g^{-1}\,\bar{D}_{a;0}$ $\displaystyle=$ $\displaystyle
g_{\Lambda}^{-1}(h_{{\alpha}_{N}^{*}}^{-1}g_{T}^{-1/2})^{2}\,{\Lambda}^{\omega_{a}^{*}}=q^{-\frac{a}{2}}\,{\Lambda}^{\omega_{a}^{*}}\,g_{\Lambda}^{-1}T^{-\omega_{a}^{*}}\,h_{{\alpha}_{N}^{*}}^{-1}g_{T}^{-1/2}h_{{\alpha}_{N}^{*}}^{-1}g_{T}^{-1/2}$
$\displaystyle=$ $\displaystyle
q^{-\frac{a}{2}}\,\bar{D}_{a,0}\,(1-{\Lambda}^{-{\alpha}_{a}^{*}})T^{-\omega_{a}^{*}}\,g^{-1}=q^{-\frac{\Lambda_{a,a}}{2}}\,\bar{D}_{a;-1}\,g^{-1}.$

For the long label $a=N$ of ${\mathfrak{g}}=C_{N}^{(1)}$,
$2\omega_{N}^{*}=\omega_{N}$, (4.11) with $n=\pm 1$ follows from

$\displaystyle g_{T}g_{\Lambda}g_{T}\,\bar{D}_{N;0}$ $\displaystyle=$
$\displaystyle
q^{\frac{N}{2}}g_{T}{\Lambda}^{\omega_{N}^{*}}g_{\Lambda}T^{\omega_{N}^{*}}\,g_{T}=q^{N}{\Lambda}^{\omega_{N}^{*}}T^{2\omega_{N}^{*}}g_{T}g_{\Lambda}g_{T}=q^{N}\bar{D}_{N;1}\,g_{T}g_{\Lambda}g_{T}$

and

$\displaystyle
g^{-1}\,\bar{D}_{N;0}=h_{{\alpha}_{N}^{*}}^{-1}g_{\Lambda}^{-1}g_{T}^{-1}g_{\Lambda}^{-1}g_{T}^{-1}{\Lambda}^{\omega_{N}^{*}}=q^{-\frac{N}{4}}\,h_{{\alpha}_{N}^{*}}^{-1}g_{\Lambda}^{-1}g_{T}^{-1}{\Lambda}^{\omega_{N}^{*}}T^{-\omega_{N}^{*}}g_{\Lambda}^{-1}g_{T}^{-1}$
$\displaystyle\quad=q^{-\frac{N}{2}}\,{\Lambda}^{\omega_{N}^{*}}\,h_{{\alpha}_{N}^{*}}^{-1}T^{-2\omega_{N}^{*}}\,(g_{\Lambda}^{-1}g_{T}^{-1})^{2}=q^{-\frac{N}{2}}\,\bar{D}_{N;0}\,(1-{\Lambda}^{-{\alpha}_{N}^{*}})T^{-2\omega_{N}^{*}}\,g^{-1}=q^{-\frac{N}{2}}\,\bar{D}_{N;-1}\,g^{-1}.$

Therefore (4.11) holds when $n=\pm 1$ and $a$ long.

Short labels. Next we verify (4.12) for short labels $a$. We claim the
following half-evolution equations hold in types
$B_{N}^{(1)},C_{N}^{(1)},A_{2N}^{(2)}$:

(4.20) $\displaystyle
q^{-\frac{1}{2}{\Lambda}_{a,a}}g_{2}\,\bar{D}_{a;0}\,g_{2}^{-1}$
$\displaystyle=$ $\displaystyle\bar{D}_{a;1},$ (4.21) $\displaystyle
q^{-\frac{1}{2}{\Lambda}_{a,a}}g_{1}\,\bar{D}_{a;1}\,g_{1}^{-1}$
$\displaystyle=$ $\displaystyle\bar{D}_{a;2}.$

To see this, Equation (4.20) for $B_{N}^{(1)}$ with $a=N$, where
$\omega_{N}^{*}=2\omega_{N}$, follows from

$g_{2}\,\bar{D}_{N;0}\,g_{2}^{-1}=g_{T}^{\frac{1}{2}}{\Lambda}^{\omega_{N}^{*}}g_{T}^{-\frac{1}{2}}=q^{\frac{N}{4}}{\Lambda}^{\omega_{N}^{*}}T^{\omega_{N}}=q^{\frac{\Lambda_{N,N}}{2}}\,\bar{D}_{N;1}.$

For the short labels of $C_{N}^{(1)},A_{2N}^{(2)}$, where
$\omega_{a}=\omega_{a}^{*}$, it follows from

$g_{2}\,\bar{D}_{a;0}\,g_{2}^{-1}=g_{T}{\Lambda}^{\omega_{a}^{*}}g_{T}^{-1}=q^{\frac{a}{2}}\,{\Lambda}^{\omega_{a}^{*}}\,T^{\omega_{a}^{*}}=q^{\frac{\Lambda_{a,a}}{2}}\,\bar{D}_{a;1}.$

Equation (4.21) ${\mathfrak{g}}=B_{N}^{(1)}$ with $a=N$, where
${\alpha}_{N}^{*}=2{\alpha}_{N}$, follows from

$\displaystyle g_{1}\,\bar{D}_{N;1}$ $\displaystyle=$ $\displaystyle
g_{T}^{\frac{1}{2}}\,{\Lambda}^{\omega_{N}^{*}}h_{{\alpha}_{N}^{*}}T^{\omega_{N}}=g_{T}^{\frac{1}{2}}{\Lambda}^{\omega_{N}^{*}}T^{\omega_{N}}(1-{\Lambda}^{-{\alpha}_{N}^{*}})\,h_{{\alpha}_{N}^{*}}$
$\displaystyle=$ $\displaystyle
q^{\frac{N}{4}}\,{\Lambda}^{\omega_{N}^{*}}T^{2\omega_{N}}(1-q{\Lambda}^{-{\alpha}_{N}^{*}}T^{-{\alpha}_{N}^{*}/2})\,g_{T}^{\frac{1}{2}}\,h_{{\alpha}_{N}^{*}}=q^{\frac{\Lambda_{N,N}}{2}}\,\bar{D}_{N;2}\,g_{1}.$

When $a\leq N-1$ for $C_{N}^{(1)},A_{2N}^{(2)}$, for $a\leq N-1$,
${\alpha}_{a}^{*}={\alpha}_{a}$ and

$\displaystyle g_{1}\,\bar{D}_{a;1}$ $\displaystyle=$ $\displaystyle
g_{T}{\Lambda}^{\omega_{a}^{*}}\,g_{\Lambda}T^{\omega_{a}}=g_{T}\,{\Lambda}^{\omega_{a}^{*}}\,T^{\omega_{a}}\,(1-{\Lambda}^{-{\alpha}_{a}^{*}})\,g_{\Lambda}$
$\displaystyle=$ $\displaystyle
q^{\frac{a}{2}}\,\bar{D}_{a;1}\,(1-{\Lambda}^{-{\alpha}_{a}^{*}}T^{-{\alpha}_{a}^{*}})\,T^{\omega_{a}}\,g_{T}\,g_{\Lambda}=q^{\frac{a}{2}}\,\bar{D}_{a;2}\,g_{1}.$

For $A_{2N}^{(2)}$, $a=N$,

$\displaystyle g_{1}\,\bar{D}_{N;1}$ $\displaystyle=$ $\displaystyle
g_{T}\,{\Lambda}^{\omega_{N}^{*}}\,g_{\Lambda}\,h_{{\alpha}_{N}^{*}}^{+}T^{\omega_{N}}=g_{T}\,{\Lambda}^{\omega_{N}^{*}}(1-q^{-\frac{1}{2}}{\Lambda}^{-{\alpha}_{N}^{*}})T^{\omega_{N}}\,g_{\Lambda}\,h_{{\alpha}_{N}^{*}}^{+}$
$\displaystyle=$ $\displaystyle
q^{\frac{N}{2}}\,\bar{D}_{N;1}\,(1-{\Lambda}^{-{\alpha}_{N}^{*}}T^{-{\alpha}_{N}^{*}})\,g_{T}\,g_{\Lambda}\,h_{{\alpha}_{N}^{*}}^{+}=q^{\frac{N}{2}}\,\bar{D}_{N;2}\,g_{1}.$

Equations (4.20),(4.21) imply (4.12) follows for $i=1,n=1$.

We also have the half-evolution equations for short labels:

(4.22) $\displaystyle
q^{\frac{1}{2}{\Lambda}_{a,a}}g_{1}^{-1}\,\bar{D}_{a;1}\,g_{1}$
$\displaystyle=$ $\displaystyle\bar{D}_{a;0},$ (4.23) $\displaystyle
q^{\frac{1}{2}{\Lambda}_{a,a}}g_{2}^{-1}\,\bar{D}_{a;0}\,g_{2}$
$\displaystyle=$ $\displaystyle\bar{D}_{a;-1}.$

Equation (4.22) is equivalent to
$\bar{D}_{a;1}=q^{-\frac{1}{2}{\Lambda}_{a,a}}g_{1}\,\bar{D}_{a;0}\,g_{1}^{-1}$,
which is unchanged if we replace $g_{1}$ by $g_{2}$, as only the $T$-dependent
part $g_{T}$ acts on $\bar{D}_{a;0}$, and the equation is therefore equivalent
to (4.20). To show (4.23) for $B_{N}^{(1)}$,

$\displaystyle g_{2}^{-1}\,\bar{D}_{N;0}$ $\displaystyle=$ $\displaystyle
g_{\Lambda}^{-1}h_{{\alpha}_{N}^{*}}^{-1}\,g_{T}^{-\frac{1}{2}}{\Lambda}^{\omega_{N}^{*}}=q^{\frac{N}{2}}\,{\Lambda}^{\omega_{N}^{*}}\,g_{\Lambda}^{-1}h_{{\alpha}_{N}^{*}}^{-1}\,T^{-\omega_{N}}\,g_{T}^{-\frac{1}{2}}$
$\displaystyle=$ $\displaystyle
q^{\frac{N}{2}}\,{\Lambda}^{\omega_{N}^{*}}\,(1-{\Lambda}^{-{\alpha}_{N}^{*}})\,T^{-\omega_{N}}g_{\Lambda}^{-1}\,h_{{\alpha}_{N}^{*}}^{-1}\,g_{T}^{-\frac{1}{2}}=q^{\frac{N}{2}}\,\bar{D}_{N;-1}\,g_{2}^{-1},$

and for $a$ short in $C_{N}^{(1)},A_{2N}^{(2)}$,

$g_{2}^{-1}\,\bar{D}_{a;0}=q^{\frac{a}{2}}\,{\Lambda}^{\omega_{a}^{*}}\,h_{{\alpha}_{N}^{*}}^{-1}g_{\Lambda}^{-1}T^{-\omega_{a}}\,g_{T}^{-1}=q^{\frac{a}{2}}\,{\Lambda}^{\omega_{a}^{*}}(1-{\Lambda}^{-{\alpha}_{a}^{*}})T^{-\omega_{a}}\,h_{{\alpha}_{N}^{*}}^{-1}g_{\Lambda}^{-1}\,g_{T}^{-1}=q^{\frac{a}{2}}\,\bar{D}_{a;-1}\,g_{2}^{-1}.$

These half-evolutions imply Equation (4.12) with $i=1,n=-1$.

The Theorem follows by induction. ∎

####  4.2.2. Integrability and conserved quantities

We claim that the time-translation operators $g$ commute with the Pieri
operators for each ${\mathfrak{g}}$, which therefore have an interpretation as
the conserved quantities of the quantum $\bar{\rm Q}$-system.

######  Theorem 4.10.

For all ${\mathfrak{g}}$, the operator $g^{({\mathfrak{g}})}({\Lambda})$
commutes with the first Pieri operator $H_{1}^{({\mathfrak{g}})}({\Lambda})$.

The proof is by explicit calculation for each ${\mathfrak{g}}$. It is given in
Appendix C.

In what follows, we need to generalize to type ${\mathfrak{g}}$ the statement
of Section 2.9 about the uniqueness of the solutions to the first Pieri
equation. Starting with
$D_{1}^{({\mathfrak{g}})}(x)\Pi_{\lambda}^{({\mathfrak{g}})}(x)={\Lambda}_{1}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)$,
writing
$\Pi_{\lambda}^{({\mathfrak{g}})}(x)=x^{\lambda}p_{\lambda}^{({\mathfrak{g}})}(x)$,
and conjugating (3.36) with $x^{-\lambda}$ results in the equation

$\left(\frac{1}{{\Lambda}_{1}}+\sum_{i=1\atop\epsilon=\pm
1}^{N}\phi_{i,\epsilon}^{({\mathfrak{g}})}(x)\left(\frac{{\Lambda}_{i}^{\epsilon}}{{\Lambda}_{1}}\Gamma_{i}^{\epsilon}-\frac{1}{{\Lambda}_{1}}\right)-1\right)\,p_{\lambda}^{({\mathfrak{g}})}(x)=0.$

The difference operator is polynomial in
$\\{{\Lambda}^{-{\alpha}_{i}^{*}}\\}$, ${\alpha}_{i}^{*}$ the simple roots of
$R^{*}$, so we may analytically continue the solution
$p_{\lambda}^{({\mathfrak{g}})}(x)$ to
$\hat{p}_{\lambda}^{({\mathfrak{g}})}(x)$, $\lambda\in{\mathbb{C}}^{N}$, and
then consider $\Lambda$ to be a formal parameter. The coefficients
$\phi_{i,\epsilon}^{({\mathfrak{g}})}(x)$ have series expansions in the
variables $x^{-{\alpha}_{i}}$, ${\alpha}_{i}$ the simple roots of $R$, hence
we may expand $\hat{p}_{\lambda}^{({\mathfrak{g}})}(x)=\sum_{\beta\in
Q_{+}(R)}c_{\beta}^{({\mathfrak{g}})}({\Lambda})x^{-\beta}$, as well as
$c_{\beta}^{({\mathfrak{g}})}({\Lambda})=\sum_{\delta\in
Q_{+}(R^{*})}c_{\beta,\delta}^{({\mathfrak{g}})}{\Lambda}^{-\delta}$. When
dealing with formal variables, we may exchange the summations and write a new
series expansion,

$\hat{p}_{\lambda}^{({\mathfrak{g}})}(x)=\sum_{\delta\in
Q_{+}(R^{*})}\hat{c}_{\delta}^{({\mathfrak{g}})}(x){\Lambda}^{-\delta},\ \
\hat{c}_{\delta}^{({\mathfrak{g}})}(x)=\sum_{\beta\in
Q_{+}(R)}c_{\beta,\delta}^{({\mathfrak{g}})}x^{-\beta}.$

The first Pieri equation (3.48) is also easily extended to formal $\lambda$,
as the dependence on ${\Lambda}^{-\alpha_{a}^{*}}$ is polynomial, while
$T_{i}:\Lambda_{j}\mapsto q^{\delta_{ij}}\Lambda_{j}$.

######  Lemma 4.11.

Let $\Theta_{\lambda}(x)$ ($\lambda$ arbitrary) be a solution of the first
Pieri equation (3.48), such that
$\Theta_{\lambda}(x)=x^{\lambda}\sum_{\beta\in
Q_{+}(R^{*})}\tau_{\beta}(x){\Lambda}^{-\beta}$. Then when $\lambda$ is
evaluated as a ${\mathfrak{g}}$-partition,

$\Pi_{\lambda}^{({\mathfrak{g}})}=\frac{\hat{c}_{0}^{({\mathfrak{g}})}(x)}{\tau_{0}(x)}\,\Theta_{\lambda}(x).$

###### Proof.

The Pieri equation is a triangular linear system for the coefficients
$\tau_{\beta}(x)$, determined uniquely for $\beta\neq 0$ up to the factor
$\tau_{0}(x)$. The same holds for the coefficients
$\hat{c}_{\beta}^{({\mathfrak{g}})}(x)$ in the expansion of
$\hat{p}_{\lambda}^{({\mathfrak{g}})}(x)$, up to
$\hat{c}_{0}^{({\mathfrak{g}})}(x)$. We deduce that
$x^{\lambda}\hat{p}_{\lambda}^{({\mathfrak{g}})}(x)=\frac{\hat{c}_{0}^{({\mathfrak{g}})}(x)}{\tau_{0}(x)}\,\Theta_{\lambda}(x)$.
The Lemma follows by specialization to ${\mathfrak{g}}$-partitions. ∎

######  Theorem 4.12.

The action of $g(\Lambda)$ on $\Pi_{\lambda}^{({\mathfrak{g}})}(x)$ is
equivalent to acting by a Gaussian in $x$:

(4.24)
$g^{({\mathfrak{g}})}({\Lambda})\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)=\gamma^{({\mathfrak{g}})}(x)\,\Pi_{\lambda}^{({\mathfrak{g}})}(x),\
\ \gamma^{({\mathfrak{g}})}(x)=\gamma(x)^{t_{1}},$

where $t_{1}=2$ if $1$ is a short label (i.e.
${\mathfrak{g}}=C_{N}^{(1)},A_{2N}^{(2)}$), and is equal to 1 otherwise.

In other words, $g^{({\mathfrak{g}})}(\Lambda)$ is the Fourier transform of
$\gamma(x)^{t_{1}}$.

###### Proof.

Like $\Pi_{\lambda}^{({\mathfrak{g}})}(x)$, the quantity
$g^{({\mathfrak{g}})}({\Lambda})\Pi_{\lambda}^{({\mathfrak{g}})}(x)$ can be
continued to arbitrary $\lambda$, and we write it as
$g^{({\mathfrak{g}})}({\Lambda})x^{\lambda}\hat{p}_{\lambda}^{({\mathfrak{g}})}(x)$.
Multiplying the first Pieri equation by $g^{({\mathfrak{g}})}$ on the left, we
get:

$g^{({\mathfrak{g}})}\,H_{1}^{({\mathfrak{g}})}({\Lambda})\,x^{\lambda}\hat{p}_{\lambda}^{({\mathfrak{g}})}(x)=H_{1}^{({\mathfrak{g}})}({\Lambda})\,\Big{(}g^{({\mathfrak{g}})}\,x^{\lambda}\hat{p}_{\lambda}^{({\mathfrak{g}})}(x)\Big{)}={\hat{e}}_{1}(x)\,\Big{(}g^{({\mathfrak{g}})}\,x^{\lambda}\hat{p}_{\lambda}^{({\mathfrak{g}})}(x)\Big{)},$

hence both
$g^{({\mathfrak{g}})}\,x^{\lambda}\hat{p}_{\lambda}^{({\mathfrak{g}})}(x)$ and
$x^{\lambda}\hat{p}_{\lambda}^{({\mathfrak{g}})}(x)$ obey the same first Pieri
rule. Apply Lemma 4.11 to the function
$\Theta_{\lambda}(x)=g^{({\mathfrak{g}})}\,x^{\lambda}\hat{p}_{\lambda}^{({\mathfrak{g}})}(x)=g^{({\mathfrak{g}})}\,x^{\lambda}(\hat{c}_{0}^{({\mathfrak{g}})}(x)+O({\Lambda}^{-{\alpha}_{i}^{*}}))$.
Expanding $g^{({\mathfrak{g}})}=1+O({\Lambda}^{-\alpha_{a}^{*}})$, we see that
the leading order term in
$g^{({\mathfrak{g}})}\,x^{\lambda}\hat{p}_{\lambda}^{({\mathfrak{g}})}(x)$ has
only contributions from the action of the $g_{T}$ parts of
$g^{({\mathfrak{g}})}$ on the leading term $x^{\lambda}$. The total
contribution of the $g_{T}$ terms is $g_{T}^{t_{1}}$, as directly read off
(4.13). Finally noting that
$(g_{T})^{t_{1}}\,x^{\lambda}=\gamma(x)^{t_{1}}\,x^{\lambda}T^{t_{1}\xi}(g_{T})^{t_{1}}$,
where $x=q^{\xi}$ (using (4.18)), we find that the leading term is
$\tau_{0}(x)=\gamma(x)^{t_{1}}\hat{c}_{0}^{({\mathfrak{g}})}(x)$, and the
Theorem follows from the relation
$\tau_{0}(x)/\hat{c}_{0}^{({\mathfrak{g}})}(x)=\gamma(x)^{t_{1}}$. ∎

######  Remark 4.13.

As noted in Remark 2.12, in the cases
${\mathfrak{g}}=D_{N+1}^{(2)},A_{2N-1}^{(2)},D_{N}^{(1)}$, the relation (4.24)
boils down to the recursion relation of [FJMM09] (Theorem 3.1) for the
coefficients $J_{\beta}^{\mu}=J_{\beta}(x=q^{\mu})$ for the root systems
$R=B_{N},C_{N},D_{N}$ respectively (see Table 1). These are the coefficients
in the series expansion
$\tilde{\Pi}^{({\mathfrak{g}})}_{\lambda}(x)=x^{\lambda}g^{({\mathfrak{g}})}_{\Lambda}\,\Pi^{({\mathfrak{g}})}_{\lambda}(x)=x^{\lambda}\sum_{\beta\in
Q_{+}}J_{\beta}(x)\Lambda^{-\beta}$, up to a rescaling $q\to q^{2}$ in the
cases ${\mathfrak{g}}=D_{N}^{(1)},D_{N+1}^{(2)}$. Here we use the notation

$g^{({\mathfrak{g}})}_{\Lambda}=g_{\Lambda}\times\left\\{\begin{matrix}&1/\left(\frac{1}{\Lambda_{N}};q^{1/2}\right)_{\infty}&{\rm
for}\ {\mathfrak{g}}=D_{N+1}^{(2)},\,R=B_{N}\\\
&1/\left(\frac{1}{\Lambda_{N}^{2}};q^{2}\right)_{\infty}&{\rm for}\
{\mathfrak{g}}=A_{2N-1}^{(2)},\,R=C_{N}\\\
&1/\left(\frac{1}{\Lambda_{N-1}\Lambda_{N}};q\right)_{\infty}&{\rm for}\
{\mathfrak{g}}=D_{N}^{(1)},\,R=D_{N}\end{matrix}\right..$

######  Corollary 4.14.

The Pieri operators $H_{a}^{({\mathfrak{g}})}({\Lambda})$, $a=1,2,...,N$ of
Eq. (3.45) are algebraically independent conserved quantities of the
${\mathfrak{g}}$-quantum $\bar{\rm Q}$-systems.

###### Proof.

To see that the time translation operator $g^{({\mathfrak{g}})}({\Lambda})$
commutes with all higher Pieri operators
$H_{a}^{({\mathfrak{g}})}({\Lambda})$, $a\in[1,N]$, act with
$(g^{({\mathfrak{g}})}({\Lambda}))^{-1}\gamma(x)^{t_{1}}$ on the Pieri
equations, and use Theorem 4.12

$g^{({\mathfrak{g}})}({\Lambda})^{-1}\gamma(x)^{t_{1}}\,H_{a}^{({\mathfrak{g}})}({\Lambda})\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)=g^{({\mathfrak{g}})}({\Lambda})^{-1}H_{a}^{({\mathfrak{g}})}({\Lambda})\,g^{({\mathfrak{g}})}({\Lambda})\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)={\hat{e}}_{a}^{({\mathfrak{g}})}(x)\,\Pi_{\lambda}^{({\mathfrak{g}})}(x),$

which implies that
$g^{({\mathfrak{g}})}({\Lambda})^{-1}H_{a}^{({\mathfrak{g}})}({\Lambda})\,g^{({\mathfrak{g}})}({\Lambda})=H_{a}^{({\mathfrak{g}})}({\Lambda})$
by the definition of the Pieri operators. The statement follows by noting that
any dependence between $H_{a}^{({\mathfrak{g}})}({\Lambda})$ would imply a
dependence between $\hat{e}_{a}^{({\mathfrak{g}})}(x)$, which are independent.
∎

###  4.3. Proof of Theorems 4.3 and 4.4

The proof of both Theorems relies on the following.

######  Theorem 4.15.

For all
${\mathfrak{g}}=D_{N}^{(1)},B_{N}^{(1)},C_{N}^{(1)},D_{N+1}^{(2)},A_{2N-1}^{(2)},A_{2N}^{(2)}$
we have the relation

(4.25)
$D_{a;n}^{({\mathfrak{g}})}(x)\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)=\bar{D}_{a;n}^{({\mathfrak{g}})}({\Lambda})\,\Pi_{\lambda}^{({\mathfrak{g}})}(x),\qquad
a\in[1,N],n\in{\mathbb{Z}}$

valid for any ${\mathfrak{g}}$-partition $\lambda$.

Theorem 4.3 follows from Theorem 4.15, as any relation satisfied by the
difference operators $\\{\bar{D}_{a;n}\\}$ implies the opposite relation for
the difference operators $\\{D_{a;n}\\}$. Theorem 4.4 is the particular case
of (4.25) with $n=0,1$. We provide a proof of (4.25) for long labels, short
labels except for $B_{N}^{(1)}$, and for the short label of $B_{N}^{(1)}$
separately.

####  4.3.1. Long labels and even time short labels

In this case, the proof is similar to that in type $A_{N-1}^{(1)}$. Acting on
both sides of the eigenvalue equation (4.7), which we write as (4.25) with
$n=0$, with $\gamma(x)^{-t_{1}n}\,(g^{({\mathfrak{g}})}({\Lambda}))^{n}$ and
using Theorem 4.12,

$\displaystyle\gamma^{-t_{1}n}\,D_{a;0}^{({\mathfrak{g}})}(x)\,(g^{({\mathfrak{g}})})^{n}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)=\gamma^{-t_{1}n}\,D_{a,0}^{({\mathfrak{g}})}(x)\,\gamma^{{t_{1}n}}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)=q^{\frac{\Lambda_{a,a}t_{a}n}{2}}\,D_{a;nt_{a}}^{({\mathfrak{g}})}(x)\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)$
$\displaystyle=(g^{({\mathfrak{g}})})^{n}\,\bar{D}_{a;0}^{({\mathfrak{g}})}({\Lambda})\,\gamma^{-t_{1}n}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)=(g^{({\mathfrak{g}})})^{n}\,\bar{D}_{a;0}^{({\mathfrak{g}})}({\Lambda})\,(g^{({\mathfrak{g}})})^{-n}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)=q^{\frac{\Lambda_{a,a}t_{a}n}{2}}\,\bar{D}_{a;nt_{a}}^{({\mathfrak{g}})}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x),$

where we have used Definition 3.15. Therefore, Eq. (4.25) holds for $n$ a
multiple of $t_{a}$.

####  4.3.2. Short labels, odd $n$ for
${\mathfrak{g}}=C_{N}^{(1)},A_{2N}^{(2)}$

In both cases, $a=1$ is a short label, $t_{1}=2$.

######  Lemma 4.16.

For ${\mathfrak{g}}=C_{N}^{(1)},A_{2N}^{(2)}$ and for any
${\mathfrak{g}}$-partition $\lambda$,

$\Big{(}D_{1,1}^{({\mathfrak{g}})}(x)-\bar{D}_{1,1}^{({\mathfrak{g}})}({\Lambda})\Big{)}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)=q^{-1}\Big{(}D_{1,-1}^{({\mathfrak{g}})}(x)-\bar{D}_{1,-1}^{({\mathfrak{g}})}({\Lambda})\Big{)}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x).$

###### Proof.

Applying Equation 3.54 with $n=0$, acting on
$\Pi_{\lambda}^{({\mathfrak{g}})}(x)$,

$\displaystyle[D_{1,0}^{({\mathfrak{g}})}(x),{\hat{e}}_{1}(x)]\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)=\left(D_{1,0}^{({\mathfrak{g}})}(x)H_{1}^{({\mathfrak{g}})}({\Lambda})-{\hat{e}}_{1}(x)\bar{D}_{1,0}^{({\mathfrak{g}})}({\Lambda})\right)\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)$
(4.26)
$\displaystyle\quad=\left[H_{1}^{({\mathfrak{g}})}({\Lambda}),\bar{D}_{1,0}^{({\mathfrak{g}})}({\Lambda})\right]\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)=(q-1)\left(D_{1,1}^{({\mathfrak{g}})}(x)-q^{-1}\,D_{1,-1}^{({\mathfrak{g}})}(x)\right)\,\Pi_{\lambda}^{({\mathfrak{g}})}(x),$

where we used the Pieri rule (3.48), and the eigenvalue equation (4.25) with
$n=0,a=1$. The only terms in $H_{1}^{({\mathfrak{g}})}({\Lambda})$ of
Equations (3.46-3.47) which fail to commute with
$\bar{D}_{1,0}^{({\mathfrak{g}})}({\Lambda})={\Lambda}_{1}$ are
$T_{1}+\left(1-\frac{{\Lambda}_{2}}{{\Lambda}_{1}}\right)T_{1}^{-1}$ if $N\geq
2$. Therefore,

$\displaystyle[H_{1}^{({\mathfrak{g}})}({\Lambda}),\bar{D}_{1,0}^{({\mathfrak{g}})}({\Lambda})]$
$\displaystyle=$
$\displaystyle[T_{1}+\left(1-\frac{{\Lambda}_{2}}{{\Lambda}_{1}}\right)T_{1}^{-1},{\Lambda}_{1}]=(q-1)\,{\Lambda}_{1}\,T_{1}+(q^{-1}-1)({\Lambda}_{1}-{\Lambda}_{2})T_{1}^{-1}$
$\displaystyle=$
$\displaystyle(q-1)\bar{D}_{1,1}^{({\mathfrak{g}})}({\Lambda})+(q^{-1}-1)\bar{D}_{1,-1}^{({\mathfrak{g}})}({\Lambda}),$

where used
$\bar{D}_{1,-1}^{({\mathfrak{g}})}({\Lambda})=({\Lambda}_{1}-{\Lambda}_{2})T_{1}^{-1}$.
The lemma follows. ∎

Defining
$\Sigma_{n}^{({\mathfrak{g}})}=(D_{1,n}^{({\mathfrak{g}})}(x)-\bar{D}_{1,n}^{({\mathfrak{g}})}({\Lambda}))\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)$,
Lemma 4.16 says that
$\Sigma_{1}^{({\mathfrak{g}})}=q^{-1}\,\Sigma_{-1}^{({\mathfrak{g}})}$.

######  Lemma 4.17.

For ${\mathfrak{g}}=C_{N}^{(1)},A_{2N}^{(2)}$, we have:

$H_{1}^{({\mathfrak{g}})}({\Lambda})\,\Sigma_{i}^{({\mathfrak{g}})}={\hat{e}}_{1}(x)\,\Sigma_{i}^{({\mathfrak{g}})},\qquad(i=\pm
1).$

###### Proof.

As $\Sigma_{-1}^{({\mathfrak{g}})}=q\,\Sigma_{1}^{({\mathfrak{g}})}$, we may
restrict ourselves to $i=1$.

(4.27) $\displaystyle
H_{1}^{({\mathfrak{g}})}({\Lambda})\,\Sigma_{1}^{({\mathfrak{g}})}$
$\displaystyle=$
$\displaystyle\Big{(}D_{1,1}^{({\mathfrak{g}})}(x)\,H_{1}^{({\mathfrak{g}})}({\Lambda})\,-H_{1}^{({\mathfrak{g}})}({\Lambda})\,\bar{D}_{1,1}^{({\mathfrak{g}})}({\Lambda})\Big{)}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)$
$\displaystyle=$
$\displaystyle\Big{\\{}{\hat{e}}_{1}(x)\,D_{1,1}^{({\mathfrak{g}})}(x)+(q-1)\big{(}D_{1,2}^{({\mathfrak{g}})}(x)-q^{-1}D_{1,0}^{({\mathfrak{g}})}(x)\big{)}$
$\displaystyle\qquad\quad-\bar{D}_{1,1}^{({\mathfrak{g}})}({\Lambda})\,H_{1}^{({\mathfrak{g}})}({\Lambda})-(q-1)\big{(}\bar{D}_{1,2}^{({\mathfrak{g}})}({\Lambda})-q^{-1}\bar{D}_{1,0}^{({\mathfrak{g}})}({\Lambda})\big{)}\Big{\\}}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)$
$\displaystyle=$
$\displaystyle\Big{\\{}{\hat{e}}_{1}(x)\,D_{1,1}^{({\mathfrak{g}})}(x)-\bar{D}_{1,1}^{({\mathfrak{g}})}({\Lambda})\,H_{1}^{({\mathfrak{g}})}({\Lambda})\Big{\\}}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)={\hat{e}}_{1}(x)\,\Sigma_{1}^{({\mathfrak{g}})},$

where we used Lemma 3.14, as well as the $a=1$, $n=0,2$ cases of (4.25) proven
above. ∎

Lemma 4.16, together with a uniqueness argument (Lemma 4.11), implies that
there exists a function ${\alpha}^{({\mathfrak{g}})}(x)$ such that
$\Sigma_{1}^{({\mathfrak{g}})}={\alpha}^{({\mathfrak{g}})}(x)\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)$,
$\Sigma_{-1}^{({\mathfrak{g}})}=q\,{\alpha}^{({\mathfrak{g}})}(x)\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)$:

$\\{D_{1,1}^{({\mathfrak{g}})}(x)-{\bar{D}}_{1,1}^{({\mathfrak{g}})}({\Lambda})-{\alpha}^{({\mathfrak{g}})}(x)\\}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)=\\{D_{1,-1}^{({\mathfrak{g}})}(x)-\bar{D}_{1,-1}^{({\mathfrak{g}})}({\Lambda})-q\,{\alpha}^{({\mathfrak{g}})}(x)\\}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)=0.$

Using this equation with $\lambda=0$,
$D_{1,-1}^{({\mathfrak{g}})}(x)\,\Pi_{0}^{({\mathfrak{g}})}(x)=D_{1,-1}^{({\mathfrak{g}})}(x)\,1=0$,
using (3.50). Similarly,
$\bar{D}_{1,-1}^{({\mathfrak{g}})}({\Lambda})\,\Pi_{\emptyset}^{({\mathfrak{g}})}(x)=({\Lambda}_{1}-{\Lambda}_{2})T_{1}^{-1}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)|_{\lambda=0}=0$,
since the prefactor vanishes. Therefore, ${\alpha}^{({\mathfrak{g}})}(x)=0$
and $\Sigma_{1}^{({\mathfrak{g}})}=\Sigma_{-1}^{({\mathfrak{g}})}=0$.
Multiplying $\Sigma_{-1}^{({\mathfrak{g}})}$ by
$(g^{({\mathfrak{g}})})^{n}\gamma^{-2n}$, we conclude that for
${\mathfrak{g}}=C_{N}^{(1)},A_{2N}^{(2)}$, $a=1$, $n\in{\mathbb{Z}}$ and any
integer partition $\lambda$, Equation (4.25) holds.

This result can be extended to all short labels for these algebras, as
follows.

######  Lemma 4.18.

Equation (4.25) holds for all short labels $a$ and $n$ odd in the case of
${\mathfrak{g}}=C_{N}^{(1)},A_{2N}^{(2)}$.

###### Proof.

Let $N_{\mathfrak{g}}$ be as in Equation (3.19). Then if $a\leq
N_{\mathfrak{g}}-1$, the relations of the quantum $\bar{\rm Q}$-system are
identical to those of type $A_{N_{\mathfrak{g}}-1}^{(1)}$, and hence we can
apply Theorem 2.8 of [DFK17b], which states that any solution of the relations
for $a=1,2,...,N_{\mathfrak{g}}-1$

$q^{a}\,\mathcal{Q}_{a;n+1}\,\mathcal{Q}_{a;n-1}=(\mathcal{Q}_{a;n})^{2}-\mathcal{Q}_{a+1;n}\mathcal{Q}_{a-1;n},\qquad\mathcal{Q}_{a;n}\mathcal{Q}_{b;n+1}=q^{{\min}(a,b)}\,\mathcal{Q}_{b;n+1}\,\mathcal{Q}_{a;n},$

satisfies

$(-1)^{a}(q-1)\mathcal{Q}_{a;n}=[\mathcal{Q}_{1;n-a+1},\mathcal{Q}_{a-1;n+1}]_{q^{a}},\qquad(a=1,2,...,N_{\mathfrak{g}}).$

Therefore the ${\mathfrak{g}}$-quantum $\bar{\rm Q}$-system solution
$\bar{D}_{a;n}^{({\mathfrak{g}})}$ obeys the opposite relations

$(-1)^{a}(q-1)\,\bar{D}_{a;n}^{({\mathfrak{g}})}=[\bar{D}_{a-1;n+1}^{({\mathfrak{g}})},\bar{D}_{1;n-a+1}^{({\mathfrak{g}})}]_{q^{a}}.$

Using this and Definition 3.16, we deduce that

$\displaystyle\\{D_{a;2n-1}^{({\mathfrak{g}})}(x)-\bar{D}_{a;2n-1}^{({\mathfrak{g}})}({\Lambda})\\}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)$
$\displaystyle\qquad\qquad=\frac{(-1)^{a}}{q-1}\\{[D_{1;2n-a}^{({\mathfrak{g}})}(x),D_{a-1;2n}^{({\mathfrak{g}})}(x)]_{q^{a}}-[\bar{D}_{a-1;2n}^{({\mathfrak{g}})}({\Lambda}),\bar{D}_{1;2n-a}^{({\mathfrak{g}})}({\Lambda})]_{q^{a}}\\}\,\Pi_{\lambda}^{({\mathfrak{g}})}(x)=0,$

by use of Eq. (4.25) for $D_{a-1;2n}^{({\mathfrak{g}})}$ and
$D_{1;2n-a}^{({\mathfrak{g}})}$ with $a$ even and for
$D_{1;2n-a}^{({\mathfrak{g}})}$ when $a$ is odd. ∎

####  4.3.3. The case of $B_{N}^{(1)}$ for odd $n$

It remains to prove (4.25) for ${\mathfrak{g}}=B_{N}^{(1)}$, $a=N$, $n$ odd.
The methods above are inapplicable, and instead we use expression (3.59) for
$D_{N;2n-1}^{(B_{N}^{(1)})}$ in terms of Rains operators.

Recall that $B_{N}^{(1)}$-partitions are both integer and half-integer
partitions, due to the spin representation with highest weight $\omega_{N}$
and character $s_{\omega_{N}}(x)=\prod_{i=1}^{N}\frac{1+x_{i}}{\sqrt{x_{i}}}$.
Denote

(4.28)
$P_{\lambda}=P_{\lambda}^{(B_{N}^{(1)})}=P_{\lambda}^{(t,-1,q^{\frac{1}{2}},-q^{\frac{1}{2}})},\qquad\tilde{P}_{\lambda}=P_{\lambda}^{(t,-q,q^{\frac{1}{2}},-q^{\frac{1}{2}})}.$

There is a factorization of Macdonald polynomials [vD95],

(4.29)
$P_{\lambda+\omega_{N}}(x)=s_{\omega_{N}}(x)\,\tilde{P}_{\lambda}(x),\qquad\hbox{$\lambda$
integer partition}.$

The parameter specialization (4.28) for $\tilde{P}_{\lambda}$ is obtained by
conjugating ${\mathcal{D}}_{1}^{(B_{N}^{(1)})}(x,q,t)$ by
$s_{\omega_{N}}^{-1}$ and identifying the resulting parameters $(a,b,c,d)$. We
must therefore consider the action of the difference operators on both
functions $\Pi_{\lambda}=\lim_{t\to\infty}P_{\lambda}$ and
$\tilde{\Pi}_{\lambda}=\lim_{t\to\infty}\tilde{P}_{\lambda}$ with $\lambda$
integer partitions.

Moreover, the Rains operators map the eigenfunctions of the $B_{N}^{(1)}$-type
difference operators to those of $B_{N}^{(1)\,^{\prime}}$-type, corresponding
to different parameters. We denote the corresponding $q$-Whittaker limit of
the eigenfunctions functions as $\Pi_{\lambda}^{\prime}$. To these parameters
there corresponds a different quantization of the Q-system, a time-translation
operator $g^{\prime}({\Lambda})$, and Hamiltonians.

Specializing to the $B_{N}^{(1)}$ parameters
$(a,b,c,d)=(t,-1,q^{\frac{1}{2}},-q^{\frac{1}{2}})$ and taking the
$t\to\infty$ limit as in (3.56), (3.8) become

$\displaystyle R_{N}^{(0)}(x)\,\Pi_{\lambda}$ $\displaystyle=$
$\displaystyle({\Lambda}_{1}{\Lambda}_{2}\cdots{\Lambda}_{N})^{\frac{1}{2}}\left(1+\frac{1}{{\Lambda}_{N}}\right)\,\Pi_{\lambda}^{\prime},$
(4.30) $\displaystyle R_{N}^{(1)}(x)\,\Pi_{\lambda}^{\prime}$ $\displaystyle=$
$\displaystyle({\Lambda}_{1}{\Lambda}_{2}\cdots{\Lambda}_{N})^{\frac{1}{2}}\,\Pi_{\lambda},$

for integer partitions $\lambda$. We identify the $B_{N}^{(1)\ ^{\prime}}$
specialization to be $(a,b,c,d)=(tq^{\frac{1}{2}},-q^{\frac{1}{2}},1,-1)$,

$\Pi_{\lambda}^{\prime}=\Pi_{\lambda}^{(B_{N}^{(1)\
^{\prime}})}=\lim_{t\to\infty}P_{\lambda}^{(tq^{\frac{1}{2}},-q^{\frac{1}{2}},1,-1)}.$

######  Lemma 4.19.

Equations (4.30) also hold for half-integer partitions.

###### Proof.

Using the specialization $(a,b,c,d)=(t,-q,q^{\frac{1}{2}},-q^{\frac{1}{2}})$
as in (4.28), in the limit $t\to\infty$ (3.8) becomes

$\tilde{R}^{(0)}_{N}\,\tilde{\Pi}_{\lambda}=q^{\frac{N}{4}}\,({\Lambda}_{1}{\Lambda}_{2}\cdots{\Lambda}_{N})^{\frac{1}{2}}\left(1+\frac{q^{-\frac{1}{2}}}{{\Lambda}_{N}}\right)\,\tilde{\Pi}_{\lambda}^{\prime},\
\
\tilde{R}^{(1)}_{N}\,\tilde{\Pi}_{\lambda}^{\prime}=q^{\frac{N}{4}}\,({\Lambda}_{1}{\Lambda}_{2}\cdots{\Lambda}_{N})^{\frac{1}{2}}\,\tilde{\Pi}_{\lambda},$

where

$\displaystyle\tilde{R}^{(0)}_{N}$ $\displaystyle=$
$\displaystyle\lim_{t\to\infty}t^{-N(N-1)/2}\,q^{-\frac{N}{4}}\mathcal{R}^{(1,-q^{\frac{1}{2}})}_{N}=s_{\omega_{N}}^{-1}\,R_{N}^{(0)}\,s_{\omega_{N}},$
$\displaystyle\tilde{R}^{(1)}_{N}$ $\displaystyle=$
$\displaystyle\lim_{t\to\infty}t^{-N(N+1)/2}\,q^{-\frac{N}{4}}\mathcal{R}^{(t,-q^{\frac{1}{2}})}_{N}=s_{\omega_{N}}^{-1}\,R_{N}^{(1)}\,s_{\omega_{N}},$

by use of the limit of (4.29). Here, $R_{N}^{(0)}$ and $R_{N}^{(1)}$ are as in
(3.56). Restoring the factors $s_{\omega_{N}}$,

$\displaystyle R_{N}^{(0)}(x)\,s_{\omega_{N}}\,\tilde{\Pi}_{\lambda}$
$\displaystyle=$
$\displaystyle(\tilde{\Lambda}_{1}\tilde{\Lambda}_{2}\cdots\tilde{\Lambda}_{N}^{\prime})^{\frac{1}{2}}\left(1+\frac{1}{\tilde{\Lambda}_{N}}\right)\,s_{\omega_{N}}\,\tilde{\Pi}_{\lambda}^{\prime},$
$\displaystyle R_{N}^{(1)}(x)\,s_{\omega_{N}}\,\tilde{\Pi}_{\lambda}^{\prime}$
$\displaystyle=$
$\displaystyle(\tilde{\Lambda}_{1}\tilde{\Lambda}_{2}\cdots\tilde{\Lambda}_{N})^{\frac{1}{2}}\,s_{\omega_{N}}\,\tilde{\Pi}_{\lambda},$

where $\tilde{\Lambda}=q^{\omega_{N}}{\Lambda}=q^{\omega_{N}+\lambda}$.
Combining this with the limit of (4.29) implies that (4.30) is satisfied for
half-integer partitions $\tilde{\lambda}=\omega_{N}+\lambda$ as well as
integer partitions. ∎

Combining equations (4.30) leads to the eigenvalue equation for
$\hat{D}^{(B_{N}^{(1)})}(x)$ of (3.56)

$\hat{D}_{N}^{(B_{N}^{(1)})}(x)\,\Pi_{\lambda}(x)=R_{N}^{(1)}(x)\,R_{N}^{(0)}(x)\,\Pi_{\lambda}={\Lambda}_{1}{\Lambda}_{2}\cdots{\Lambda}_{N}\left(1+\frac{1}{{\Lambda}_{N}}\right)\,\Pi_{\lambda},$

which is consistent with the relation
$\hat{D}_{N}^{(B_{N}^{(1)})}=R_{N}^{(1)}\,R_{N}^{(0)}=D_{N,0}+D_{N-1,0}$ and
the eigenvalues $\bar{D}_{a,0}$ in Definition (4.6).

The $q$-Whittaker limit of the difference operators for the
${B_{N}^{(1)}}^{\prime}$ parameters are obtained as in (A.38), and are denoted
by $D_{a}^{\prime}(x)=D_{a}^{(B_{N}^{(1)\,^{\prime}})}(x)$, $a=1,2,...,N$. The
corresponding limit of the Rains operator (3.9) is
$\hat{D}_{N}^{\prime}(x)=R_{N}^{(0)}(x)\,R_{N}^{(1)}(x)=D_{N}^{\prime}(x)+D_{N-1}^{\prime}(x)$.
This identification follows from the $t\to\infty$ limit of Lemma A.10, and
from the eigenvalues, which are identical to those of the $B_{N}^{(1)}$ case,
since $\xi_{\mathfrak{g}}=\xi_{{\mathfrak{g}}^{\prime}}$. As in the
$B_{N}^{(1)}$ case, We define the generalized Macdonald operators

$\displaystyle D_{a;n}^{\prime}(x)$ $\displaystyle=$ $\displaystyle
q^{-na/2}\,\gamma^{-n}\,D_{a}^{\prime}(x)\,\gamma^{n},\qquad(a=1,2,...,N-1),$
$\displaystyle D_{N;2n}^{\prime}(x)$ $\displaystyle=$ $\displaystyle
q^{-nN/2}\,\gamma^{-n}\,D_{N}^{\prime}(x)\,\gamma^{n},$ (4.31) $\displaystyle
D_{N;2n+1}^{\prime}(x)$ $\displaystyle=$ $\displaystyle
q^{-nN/4}\,R_{N;n+1}^{(0)}(x)\,R_{N;n}^{(1)}(x)=q^{-nN/2}\,\gamma^{-n}\,D_{N;1}^{\prime}(x)\,\gamma^{n}.$

Note the reversal of the order in the last product and the different time
index compared to (3.59). The operators $D_{a;n}^{\prime}$ will be shown to
satisfy the set of recursion relations below.

######  Definition 4.20.

The type ${B_{N}^{(1)}}^{\prime}$-quantum Q-system relations666This “quantum
Q-system” is new, and we call it a Q-system by analogy with the other
${\mathfrak{g}}$ cases.  are the same as the type $B_{N}^{(1)}$ relations in
(4.3)–(4.5) except for the two equations with $a=N$:

$\displaystyle q^{N/2}{\mathcal{Q}}_{N;2n+2}\,{\mathcal{Q}}_{N;2n}$
$\displaystyle=$
$\displaystyle{\mathcal{Q}}_{N;2n+1}^{2}-q^{\frac{N-1}{2}-n}\,{\mathcal{Q}}_{N-1;n+1}\,{\mathcal{Q}}_{N-1;n},$
$\displaystyle q^{N/2}{\mathcal{Q}}_{N;2n+1}\,{\mathcal{Q}}_{N;2n-1}$
$\displaystyle=$
$\displaystyle({\mathcal{Q}}_{N;2n}-q^{-\frac{n}{2}}\,{\mathcal{Q}}_{N-1;n})({\mathcal{Q}}_{N;2n}+q^{\frac{1-n}{2}}\,{\mathcal{Q}}_{N-1;n}).$

In particular, the q-commutation relations are as in type $B_{N}^{(1)}$ (4.3).

The eigenvalue equation for $D_{a}^{\prime}$ corresponding to any $\lambda$ is
$D_{a;0}^{\prime}(x)\,\Pi_{\lambda}^{\prime}={\Lambda}^{\omega_{a}^{*}}\,\Pi_{\lambda}^{\prime}$,
with $\omega_{a}^{*}$ a fundamental weight of type $C_{N}$. Therefore, the
candidate Fourier transforms $\bar{D}_{a,n}^{\prime}(\Lambda)$ are defined so
that they satisfy the opposite quantum Q-system to that of Definition 4.20,
subject to the same initial data as in type $B_{N}^{(1)}$,
$\bar{D}^{\prime}_{a;0}={\Lambda}^{\omega_{a}^{*}}$ and
$\bar{D}_{a;1}^{\prime}={\Lambda}^{\omega_{a}^{*}}T^{\omega_{a}}$,
$\bar{D}_{N;1}^{\prime}={\Lambda}^{\omega_{N}^{*}}T^{\omega_{N}}$. Together
with the recursions of Def. 4.20, this determines $\bar{D}_{a;n}^{\prime}$ for
all $a,n$. In particular,

$\displaystyle\bar{D}_{a;-1}^{\prime}$ $\displaystyle=$
$\displaystyle\bar{D}_{a;0}^{\prime}\left(1-\frac{{\Lambda}_{a+1}}{{\Lambda}_{a}}\right)T^{-\omega_{a}}\qquad(a=1,2,...,N-1),$
$\displaystyle\bar{D}_{N;-1}^{\prime}$ $\displaystyle=$
$\displaystyle\bar{D}_{N;0}^{\prime}\left(1-\frac{1}{{\Lambda}_{N}}\right)\left(1+\frac{q^{\frac{1}{2}}}{{\Lambda}_{N}}\right)T^{-\omega_{N}},\
\
\bar{D}_{N;2}^{\prime}=\bar{D}_{N;1}^{\prime}\left(1-q^{\frac{1}{2}}\frac{1}{{\Lambda}_{N}^{2}}\frac{1}{T_{N}}\right)T^{\omega_{N}}.$

As before, these are sufficient to determine the form of the time-translation
operator $g^{\prime}(\Lambda)$.

######  Lemma 4.21.

The operator

(4.32) $\displaystyle g^{\prime}$ $\displaystyle=$ $\displaystyle
g_{T}^{\frac{1}{2}}\,g_{{\Lambda}_{N}}^{\prime}\,g_{T}^{\frac{1}{2}}\,g_{{\Lambda}_{N}}^{\prime\prime}\,g_{\Lambda},$
$\displaystyle g_{{\Lambda}_{N}}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{(q^{\frac{1}{2}}{\Lambda}_{N}^{-2};q)_{\infty}},\quad
g_{{\Lambda}_{N}}^{\prime\prime}=\frac{1}{({\Lambda}_{N}^{-1};q^{\frac{1}{2}})_{\infty}\,(-q^{\frac{1}{2}}{\Lambda}_{N}^{-1};q^{\frac{1}{2}})_{\infty}},$

where $g_{T}$ is as in (2.23) and $g_{\Lambda}$ as in (2.26), is the time
translation operator for the $B_{N}^{(1)\,^{\prime}}$ opposite quantum
Q-system. That is, for all $n\in{\mathbb{Z}}$,

$\displaystyle\bar{D}_{a;n}^{\prime}$ $\displaystyle=$ $\displaystyle
q^{-an/2}\,(g^{\prime})^{n}\,\bar{D}_{a}^{\prime}\,(g^{\prime})^{-n}\qquad(a=1,2,...,N-1),$
$\displaystyle\bar{D}_{N;2n+i}^{\prime}$ $\displaystyle=$ $\displaystyle
q^{-Nn/2}(g^{\prime})^{n}\,\bar{D}_{N;i}^{\prime}\,(g^{\prime})^{-n},\quad
i=0,1.$

The Pieri rules for $B_{N}^{(1)\,^{\prime}}$ are obtained by duality. Using
Theorem B.1 with $(a,b,c,d)=(1,-1,tq^{\frac{1}{2}},-q^{\frac{1}{2}})$,
$(a^{*},b^{*},c^{*},d^{*})=(t^{\frac{1}{2}},-t^{-\frac{1}{2}},t^{\frac{1}{2}}q^{\frac{1}{2}},-t^{-\frac{1}{2}}q^{\frac{1}{2}})$,
the $q$-Whittaker limit of the first Pieri operator is

$\displaystyle
H_{1}^{(B_{N}^{(1)\,^{\prime}})}({\Lambda})=T_{1}+\sum_{a=2}^{N}\left(1-\frac{{\Lambda}_{a}}{{\Lambda}_{a-1}}\right)T_{a}+\sum_{a=1}^{N-1}\left(1-\frac{{\Lambda}_{a+1}}{{\Lambda}_{a}}\right)T_{a}^{-1}$
$\displaystyle+\left(1-\frac{1}{{\Lambda}_{N}}\right)\left(1+\frac{q}{{\Lambda}_{N}}\right)\left(1-\frac{q}{{\Lambda}_{N}^{2}}\right)T_{N}^{-1}+\frac{q^{\frac{1}{2}}-q^{-\frac{1}{2}}}{{\Lambda}_{N}}-\frac{q^{\frac{1}{2}}+q^{-\frac{1}{2}}}{{\Lambda}_{N}^{2}}+\frac{q^{-\frac{1}{2}}}{{\Lambda}_{N-1}{\Lambda}_{N}}.$

By direct calculation, this operator commutes with $g^{\prime}(\Lambda)$ and
as a consequence

(4.33)
$g^{\prime}({\Lambda})\,\Pi_{\lambda}^{\prime}(x)=\gamma(x)\,\Pi_{\lambda}^{\prime}(x)\
.$

We want to show that
$D_{N;1}(x)\Pi_{\lambda}=\bar{D}_{N;1}(\Lambda)\Pi_{\lambda}$ (case $a=N,n=1$
of (4.25)). Using 4.12, Definition (3.59) and the relations (4.30), we find

(4.34) $\displaystyle D_{N;1}(x)\,\Pi_{\lambda}$ $\displaystyle=$
$\displaystyle q^{-\frac{N}{4}}\,R_{N;0}^{(1)}\,R_{N;1}^{(0)}\,\Pi_{\lambda}$
$\displaystyle=$ $\displaystyle
q^{-\frac{3N}{8}}\,g\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}\left(1+\frac{1}{{\Lambda}_{N}}\right)\,(g^{\prime})^{-1}\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}\,\Pi_{\lambda}.$

Recall the notation
$h_{{\alpha}_{N}^{*}}=\prod_{n=0}^{\infty}\left(1-\frac{q^{n}}{{\Lambda}_{N}^{2}}\right)^{-1}$,
then we have the relations
$\left(1+\frac{1}{{\Lambda}_{N}}\right)(g_{{\Lambda}_{N}}^{\prime\prime})^{-1}=h_{{\alpha}_{N}^{*}}^{-1}$
and
$T_{N}^{\frac{1}{4}}(g_{{\Lambda}_{N}}^{\prime})^{-1}=h_{{\alpha}_{N}^{*}}^{-1}T_{N}^{\frac{1}{4}}$.
These allow to compute

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!g\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}\left(1+\frac{1}{{\Lambda}_{N}}\right)\,(g^{\prime})^{-1}\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle
g_{T}^{\frac{1}{2}}\,h_{{\alpha}_{N}^{*}}\,g_{T}^{\frac{1}{2}}\,h_{{\alpha}_{N}^{*}}\,g_{\Lambda}({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}\\!\\!\left(1+\frac{1}{{\Lambda}_{N}}\right)g_{\Lambda}^{-1}\,(g_{{\Lambda}_{N}}^{\prime\prime})^{-1}g_{T}^{-\frac{1}{2}}\,(g_{{\Lambda}_{N}}^{\prime})^{-1}g_{T}^{-\frac{1}{2}}\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle
g_{T}^{\frac{1}{2}}\,h_{{\alpha}_{N}^{*}}\,g_{T}^{\frac{1}{2}}\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}g_{T}^{-\frac{1}{2}}\,(g_{{\Lambda}_{N}}^{\prime})^{-1}g_{T}^{-\frac{1}{2}}\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle
q^{\frac{N}{16}}g_{T}^{\frac{1}{2}}\,h_{{\alpha}_{N}^{*}}\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}(T_{1}\cdots
T_{N})^{\frac{1}{4}}\,(g_{{\Lambda}_{N}}^{\prime})^{-1}\,g_{T}^{-\frac{1}{2}}\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle
q^{\frac{N}{16}}g_{T}^{\frac{1}{2}}\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}(T_{1}\cdots
T_{N})^{\frac{1}{4}}\,g_{T}^{-\frac{1}{2}}\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle
q^{\frac{N}{8}}({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}\,(T_{1}\cdots
T_{N})^{\frac{1}{2}}\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}$ (4.35)
$\displaystyle=$ $\displaystyle
q^{\frac{3N}{8}}\,{\Lambda}_{1}\cdots{\Lambda}_{N}\,(T_{1}\cdots
T_{N})^{\frac{1}{2}}=q^{\frac{3N}{8}}\,\bar{D}_{N;1}.$

Combining this with (4.34) results in the relation
$D_{N;1}(x)\Pi_{\lambda}=\bar{D}_{N;1}(\Lambda)\Pi_{\lambda}$. Multiplying by
$\gamma^{-n}g^{n}$ and using (4.33):

(4.36)
$D_{N;2n+1}^{(B_{N}^{(1)})}(x)\,\Pi_{\lambda}^{(B_{N}^{(1)})}(x)=\bar{D}_{N;2n+1}^{(B_{N}^{(1)})}({\Lambda})\,\Pi_{\lambda}^{(B_{N}^{(1)})}(x),\qquad(n\in{\mathbb{Z}}).$

This completes the proof of (4.25), and Theorems 4.3 and 4.4 follow.

######  Corollary 4.22.

The $D^{\prime}$-operators satisfy the quantum
$B_{N}^{(1)\,^{\prime}}$-quantum Q-system of Definition 4.20. Moreover we have
the raising operator conditions for any $B_{N}^{(1)}$-partition $\lambda$:

$D_{a;1}^{\prime}(x)\,\Pi_{\lambda}^{\prime}={\Lambda}^{\omega_{a}^{*}}\,\Pi_{\lambda+\omega_{a}}^{\prime},\qquad(a\in[1,N]),$

$\omega_{a},\omega_{a}^{*}$ the fundamental weights of $B_{N},C_{N}$.

###### Proof.

Starting from the eigenvalue equations
$D^{\prime}_{a}(x)\,\Pi_{\lambda}^{\prime}=\bar{D}_{a}^{\prime}\,\Pi_{\lambda}^{\prime}$
for $a=1,2,...,N$ and multiplying with $\gamma^{-n}(g^{\prime})^{n}$, and
still denoting $\bar{X}$ the prime Fourier transform of $X$,

$D^{\prime}_{a;t_{a}n}(x)\Pi_{\lambda}^{\prime}=q^{-na/2}\gamma^{-n}\,D^{\prime}_{a}(x)\,\gamma^{n}\,\Pi_{\lambda}^{\prime}=q^{-na/2}(g^{\prime})^{n}\,\bar{D}^{\prime}_{a}\,(g^{\prime})^{-n}\,\Pi_{\lambda}^{\prime}=\bar{D}^{\prime}_{a;t_{a}n}\,\Pi_{\lambda}^{\prime}.$

For the short label $a=N$, using (4.31), (4.30), (4.35) and
$\bar{D}_{N;1}=\bar{D}_{N;1}^{\prime}={\Lambda}^{\omega_{N}^{*}}T^{\omega_{N}}$:

$\displaystyle D^{\prime}_{N;1}(x)\,\Pi_{\lambda}^{\prime}$ $\displaystyle=$
$\displaystyle
R_{N;1}^{(0)}\,R_{N}^{(1)}\,\Pi_{\lambda}^{\prime}=q^{-\frac{N}{8}}\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}g\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}\left(1+\frac{1}{{\Lambda}_{N}}\right)\,(g^{\prime})^{-1}\,\Pi_{\lambda}$
$\displaystyle=$ $\displaystyle
q^{-\frac{N}{8}}\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{\frac{1}{2}}\,q^{\frac{3N}{8}}\,\bar{D}_{N;1}\,({\Lambda}_{1}\cdots{\Lambda}_{N})^{-\frac{1}{2}}\,\Pi_{\lambda}=\bar{D}_{N;1}^{\prime}\,\Pi_{\lambda}^{\prime}.$

The relation is easily extended by multiplication with
$\gamma^{-n}(g^{\prime})^{n}$, resulting in
$D^{\prime}_{N;2n+1}(x)\,\Pi_{\lambda}^{\prime}=\bar{D}_{N;2n+1}^{\prime}\,\Pi_{\lambda}^{\prime}$
for all $n\in{\mathbb{Z}}$. The Corollary follows.

∎

##  5\. Universal Solutions

The relations between Macdonald eigenvalue equations and Pieri rules are
embodied by the duality property of Macdonald polynomials (3.28). We now
discuss a reformulation of this duality in terms of universal solutions along
the same lines as the case of type $A$, treated in Section 2.13. This allows
to re-prove the main results of this paper in terms of universal solutions.

###  5.1. Universal solutions

####  5.1.1. Universal Koornwinder-Macdonald and ${\mathfrak{g}}$-Macdonald
eigenvalue solutions

As in the case of type A, it is extremely fruitful to think directly of $x$
and $s$ as dual variables, whose roles may be interchanged. To this end, we
use changes of variables:

$x_{i}=q^{\mu_{i}}\,t^{{\rho_{i}}^{*}}=q^{\mu_{i}}\,at^{N-i},\quad
s_{i}=q^{\lambda_{i}}\,t^{\rho_{i}}=q^{\lambda_{i}}\,\sigma
t^{N-i},\qquad\lambda,\mu\in{\mathbb{C}}^{N},$

where we use the notations of Section 3.1.4, see Equation (3.13). Note that if
we write $P_{\lambda}^{(a,b,c,d)}(x)=x^{\lambda}p_{\lambda}^{(a,b,c,d)}(x)$
and substitute this into the Koornwinder-Macdonald eigenvalue equation (3.5),
we obtain

$\left(\frac{(1-t^{N})(1+\sigma t^{N-1})}{1-t}-\sigma
t^{N-1}\hat{e}_{1}(s)+\sum_{i=1\atop\epsilon=\pm
1}^{N}\Phi_{i,\epsilon}^{(a,b,c,d)}(x)({\Lambda}_{i}^{\epsilon}\Gamma_{i}^{\epsilon}-1)\right)\,p_{\lambda}^{(a,b,c,d)}(x;s)=0,$

where we have conjugated (3.4) with $x^{-\lambda}$. Noting that
$\Phi_{i,\epsilon}^{(a,b,c,d)}(x)$ can be expanded in series of the variables
$x^{-{\alpha}_{i}}$, ${\alpha}_{i}$ the $B_{N}$ simple roots, this suggests
what we call a universal solution $P^{(a,b,c,d)}(s;x)$ of the Koornwinder-
Macdonald eigenvalue equation (3.5) in the form

(5.1) $P^{(a,b,c,d)}(x;s)=q^{\lambda\cdot\mu}\,\sum_{\beta\in
Q_{+}}c_{\beta}^{(a,b,c,d)}(s)\,x^{-\beta},\ \ c_{0}^{(a,b,c,d)}(s)=1,$

where $Q_{+}$ denotes the positive cone of the root lattice of $B_{N}$. The
normalizing prefactor for $P^{(a,b,c,d)}(x;s)$ is such that
$q^{\lambda\cdot\mu}=x^{\lambda}\,t^{-\rho^{*}\cdot\lambda}=s^{\mu}\,t^{-\rho\cdot\mu}$,
and is invariant under the interchange of $\lambda\leftrightarrow\mu$,
therefore under $x\leftrightarrow s$. Equation (3.5) for $P^{(a,b,c,d)}(x;s)$
and generic $s$,

(5.2) ${\mathcal{D}}_{1}^{(a,b,c,d)}\,P^{(a,b,c,d)}(x;s)=\sigma
t^{N-1}\,\hat{e}_{1}(s)\,P^{(a,b,c,d)}(x;s),$

is equivalent to a linear triangular, generically nonsingular, system for the
coefficients $c_{\beta}^{(a,b,c,d)}(s)$, which uniquely fixes them for all
$\beta\in Q_{+}$. The solution $P^{(a,b,c,d)}(x;s)$ is therefore unique, and
we refer to it as the universal Koornwinder-Macdonald solution. The
normalization ${c}_{0}^{(a,b,c,d)}(s)$ uniquely fixes the solution, which is
otherwise determined up to the overall normalization given by this
coefficient.

Specializing the Koornwinder parameters $(a,b,c,d)$ according to Table 1, we
obtain the universal solutions $P^{({\mathfrak{g}})}(x;s)$ of the first
${\mathfrak{g}}$-Macdonald eigenvalue equations (3.23):

(5.3)
${\mathcal{D}}_{1}^{({\mathfrak{g}})}\,P^{({\mathfrak{g}})}(x;s)=t^{\rho_{1}}\,\hat{e}_{1}(s)\,P^{({\mathfrak{g}})}(x;s).$

These universal solutions have the expansion

(5.4) $P^{({\mathfrak{g}})}(x;s)=q^{\mu\cdot\lambda}\,\sum_{\beta\in
Q_{+}(R)}c_{\beta}^{({\mathfrak{g}})}(s)\,x^{-\beta},\ \
c_{0}^{({\mathfrak{g}})}(s)=1,$

with the coefficients $c_{\beta}^{({\mathfrak{g}})}(s)$ uniquely determined by
(5.3), up to the normalization $c_{0}^{({\mathfrak{g}})}(s)$.

######  Remark 5.1.

The apparent discrepancy between $Q_{+}$ in (5.1) and $Q_{+}(R)$ in (5.4) is
just an artifact of the specializations. For instance using the $C_{N}^{(1)}$
specialization $(a,b,c,d)=t^{1/2}(1,-1,q^{1/2},-q^{1/2})$ leads to an operator
${\mathcal{D}}_{1}^{(C_{N})}(x;q,t)$ which has an expansion as a series of the
variables $\\{x_{i+1}/x_{i}\\}_{1\leq i\leq N-1}$ and $x_{N}^{-2}$, as opposed
to the generic case ${\mathcal{D}}_{1}^{(a,b,c,d)}(x;q,t)$ which has an
expansion as a series of the variables $\\{x_{i+1}/x_{i}\\}_{1\leq i\leq N-1}$
and $x_{N}^{-1}$. This simply means that the series solution of the Macdonald
eigenvalue equation at the specific $C_{N}^{(1)}$ specialization is an even
function of $x_{N}^{-1}$, i.e. the coefficients of odd powers of $x_{N}^{-1}$
vanish.

######  Remark 5.2.

Specializing $s=q^{\lambda}t^{\rho}$ to $\lambda$ an integer partition, and
using the uniqueness of the universal solution, we recover the Koornwinder
polynomial

(5.5)
$P^{(a,b,c,d)}(x;s=q^{\lambda}t^{\rho})=t^{-\rho^{*}\cdot\lambda}\,P_{\lambda}^{(a,b,c,d)}(x).$

The specialization therefore truncates the series (5.1) to finitely many
terms. Specializing the Koornwinder parameters to the values in Table 1,

(5.6)
$P^{({\mathfrak{g}})}(x;s=q^{\lambda}t^{\rho})=t^{-\rho^{*}\cdot\lambda}\,P_{\lambda}^{({\mathfrak{g}})}(x),$

for $\lambda$ an integer partition, $\rho=\rho^{({\mathfrak{g}})}$ and
$\rho^{*}=\rho^{({\mathfrak{g}}^{*})}$. Moreover, by uniqueness, (5.6) holds
also for $\lambda$ any ${\mathfrak{g}}$-partition, for example the non-integer
partitions in the cases
${\mathfrak{g}}=D_{N}^{(1)},B_{N}^{(1)},D_{N+1}^{(2)}$. This presentation is
therefore more economical, as a single universal function $P^{(a,b,c,d)}(x;s)$
contains the information on all the ${\mathfrak{g}}$-Macdonald polynomials as
well.

Another important specialization of the universal Koornwinder function
$P^{(a,b,c,d)}(x;s)$ is the Koornwinder Baker-Akhiezer quasi-polynomials
introduced by Chalykh [Cha02]. The latter correspond to specializing the
paramaters $a,b,c,d,t$ to arbitrary negative integer powers of $q$. The effect
is again a truncation of the series to finitely many terms.

####  5.1.2. Universal Pieri solutions

We also define the universal solution $Q^{(a,b,c,d)}(s;x)$ of the first Pieri
equation (3.17):

(5.7) $Q^{(a,b,c,d)}(s;x)=q^{\lambda\cdot\mu}\,\sum_{\beta\in
Q_{+}}{\bar{c}}_{\beta}^{(a,b,c,d)}(x)\,s^{-\beta},\ \
{\bar{c}}_{0}^{(a,b,c,d)}(x)=1$

subject to:

${\hat{e}}_{1}(x)\,Q^{(a,b,c,d)}(s;x)=\hat{\mathcal{H}}_{1}^{(a,b,c,d)}(s;q,t)\,Q^{(a,b,c,d)}(s;x),$

where

(5.8) $\displaystyle\hat{\mathcal{H}}_{1}^{(a,b,c,d)}(s;q,t)$
$\displaystyle:=$ $\displaystyle
t^{-\rho^{*}\cdot\lambda}\,{\mathcal{H}}_{1}^{(a,b,c,d)}(s;q,t)\,t^{\rho^{*}\cdot\lambda}$
$\displaystyle=$
$\displaystyle\frac{1}{at^{N-1}}\,\Delta^{(a^{*},b^{*},c^{*},d^{*})}(s)^{-1}\,{\mathcal{D}}_{1}^{(a^{*},b^{*},c^{*},d^{*})}(s;q,t)\,\Delta^{(a^{*},b^{*},c^{*},d^{*})}(s).$

The series expansion (5.7) exists because
$\hat{\mathcal{H}}_{1}^{(a,b,c,d)}(s;q,t)$ has a series expansion in the
variables $s^{-{\alpha}_{i}}$. As above, the normalization
${\bar{c}}_{0}^{(a,b,c,d)}(x)=1$ uniquely fixes the solution.

Specializing the parameters $(a,b,c,d)$ as in Table 1, we have the universal
solutions to the first ${\mathfrak{g}}$-Pieri equations (3.32):

(5.9)
${\hat{e}}_{1}(x)\,Q^{({\mathfrak{g}})}(s;x)=\hat{\mathcal{H}}_{1}^{({\mathfrak{g}})}(s;q,t)\,Q^{({\mathfrak{g}})}(s;x),$

where we have used the fact that ${\hat{e}}_{1}^{(R)}(x)={\hat{e}}_{1}(x)$ for
all $R$, and $\hat{\mathcal{H}}_{1}^{({\mathfrak{g}})}(s=\Lambda
t^{\rho};q,t):=t^{-\rho^{*}\cdot\lambda}\,{\mathcal{H}}_{1}^{({\mathfrak{g}})}(\Lambda;q,t)\,t^{\rho^{*}\cdot\lambda}$,
with ${\mathcal{H}}_{1}^{({\mathfrak{g}})}(\Lambda;q,t)$ as in (3.33). The
solution $Q^{({\mathfrak{g}})}(s;x)$ is a series of the form

$Q^{({\mathfrak{g}})}(s;x)=q^{\mu\cdot\lambda}\sum_{\beta\in
Q_{+}^{*}}{\bar{c}}_{\beta}^{({\mathfrak{g}})}(x)\,s^{-\beta},\ \
{\bar{c}}_{0}^{({\mathfrak{g}})}(x)=1,$

where the sum extends over the positive root cone $Q_{+}^{*}$ of $R^{*}$. Due
to triangularity, the coefficients of the series are uniquely determined by
(5.9).

###  5.2. Duality

####  5.2.1. Duality of universal functions

The duality of Macdonald polynomials can be extended to the universal
functions as follows.

######  Theorem 5.3.

The functions $Q^{(a,b,c,d)}(s;x)$ and $P^{(a,b,c,d)}(x;s)$ are related via

(5.10) $Q^{(a,b,c,d)}(s;x)=\frac{P^{(a,b,c,d)}(x;s)}{\Delta^{(a,b,c,d)}(x)},$

with $\Delta^{(a,b,c,d)}$ as in (3.14).

###### Proof.

The proof is similar to that of Theorem 2.15. The universal solution
$P^{(a,b,c,d)}(x;s)$ also obeys the Pieri equation (5.8), as a consequence of
the existence of a solution to the bispectral problem [Che95b, vD96, Sah99,
Cha02], i.e. of both Koornwinder eigenvalue and Pieri equations, and of the
uniqueness of the solution up to the overall normalization determined by the
leading coefficient. Expanding the coefficients
$c_{\beta}^{(a,b,c,d)}(s)=\sum_{\delta\in
Q_{+}}c_{\beta,\delta}^{(a,b,c,d)}s^{-\delta}$ allows to write an expansion

$P^{(a,b,c,d)}(x;s)=q^{\lambda\cdot\mu}\sum_{\delta\in
Q_{+}}\hat{c}_{\delta}^{(a,b,c,d)}(x)\,s^{-\delta},\ \
\hat{c}_{\delta}^{(a,b,c,d)}(x)=\sum_{\beta\in
Q_{+}}c_{\beta,\delta}^{(a,b,c,d)}x^{-\beta}.$

Ehe Pieri equation uniquely fixes the coefficients in this expansion, up to
the overall factor $\hat{c}_{0}^{(a,b,c,d)}(x)$, hence
$P^{(a,b,c,d)}(x;s)=\hat{c}_{0}^{(a,b,c,d)}(x)\,Q^{(a,b,c,d)}(s;x)$ as a
series in $s^{-{\alpha}_{i}}$.

To compute $\hat{c}_{0}^{(a,b,c,d)}(x)$, we note that it can be extracted as
the successive limits $s_{1}\to\infty$, $s_{2}\to\infty$, …, $s_{N}\to\infty$
of $P^{(a,b,c,d)}(x;s)$. Let us examine the $m$ eigenvalue equation (3.6), for
$P^{(a,b,c,d)}(x;s)=q^{\lambda.\mu}(\hat{c}_{0}^{(a,b,c,d)}(x)+O({\Lambda}^{-{\alpha}_{i}}))$.
Writing

${\mathcal{D}}_{m}^{(a,b,c,d)}=d_{m}^{(a,b,c,d)}(x)+\sum_{I\subset[1,N]\atop|I|\leq
m}\sum_{\epsilon_{k}=\pm 1\atop k\in I}d_{I,\epsilon,m}(x)\prod_{k\in
I}\Gamma_{k}^{\epsilon_{k}},$

we have

$\left(d_{m}^{(a,b,c,d)}(x)-\theta_{m}\,\hat{e}_{m}(x)+\sum_{I\subset[1,N]\atop|I|\leq
m}\sum_{\epsilon_{k}=\pm 1\atop k\in I}d_{I,\epsilon,m}(x)\prod_{k\in
I}{\Lambda}_{k}^{\epsilon_{k}}\,\Gamma_{k}^{\epsilon_{k}}\right)(\hat{c}_{0}^{(a,b,c,d)}(x)+O({\Lambda}^{-{\alpha}_{i}}))=0,$

where $\theta_{m}=\sigma^{m}t^{m(N-\frac{m+1}{2})}$. Dividing by
${\Lambda}_{1}{\Lambda}_{2}\cdots{\Lambda}_{m}$, we find that all the terms
have a factor of the form ${\Lambda}^{-\beta}$ for some $\beta\in Q_{+}$, and
only those with $\beta=0$ survive in the limit
$|{\Lambda}_{1}|>>|{\Lambda}_{2}|>>\cdots>>|{\Lambda}_{m}|>>1$. This leaves us
with

(5.11)
$\left(\Gamma_{1}\Gamma_{2}\cdots\Gamma_{m}-\frac{\theta_{m}^{2}}{d_{[1,m],(+)^{m},m}(x)}\right)\hat{c}_{0}^{(a,b,c,d)}(x)=0.$

Let us compute $d_{[1,m],(+)^{m},m}(x)$. From Definition A.8, and the explicit
expression of the van Diejen operators (A.8-A.9), we easily see that the
coefficient of $\Gamma_{1}\Gamma_{2}\cdots\Gamma_{m}$ in
${\mathcal{D}}_{m}^{(a,b,c,d)}$ comes from the highest order van Diejen
operator ${\mathcal{V}}_{m}^{(a,b,c,d)}$, and more precisely from the term
$s=1$, $J_{1}=J=\\{1,2,...,m\\}$ in (A.8):

$\displaystyle d_{[1,m],(+)^{m},m}(x)$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{m}\frac{(1-ax_{i})(1-bx_{i})(1-cx_{i})(1-dx_{i})}{(1-x_{i}^{2})(1-qx_{i}^{2})}\prod_{1\leq
i<j\leq
m}\frac{1-tx_{i}x_{j}}{1-x_{i}x_{j}}\frac{1-qtx_{i}x_{j}}{1-qx_{i}x_{j}}$
$\displaystyle\qquad\times\prod_{1\leq i\leq m<j\leq
N}\frac{1-tx_{i}x_{j}}{1-x_{i}x_{j}}\frac{tx_{i}-x_{j}}{x_{i}-x_{j}}.$

Using

$\displaystyle\frac{\theta_{m}^{2}}{d_{[1,m],(+)^{m},m}(x)}$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{m}\frac{(1-\frac{1}{x_{i}^{2}})(1-\frac{1}{qx_{i}^{2}})}{(1-\frac{1}{ax_{i}})(1-\frac{1}{bx_{i}})(1-\frac{1}{cx_{i}})(1-\frac{1}{dx_{i}})}\prod_{1\leq
i<j\leq
m}\frac{1-\frac{1}{x_{i}x_{j}}}{1-\frac{1}{tx_{i}x_{j}}}\frac{1-\frac{1}{qx_{i}x_{j}}}{1-\frac{1}{qtx_{i}x_{j}}}$
$\displaystyle\qquad\times\prod_{1\leq i\leq m<j\leq
N}\frac{1-\frac{1}{x_{i}x_{j}}}{1-\frac{1}{tx_{i}x_{j}}}\frac{1-\frac{x_{j}}{x_{i}}}{1-\frac{x_{j}}{tx_{i}}}$

we conclude that $\hat{c}_{0}^{(a,b,c,d)}(x)$ and $\Delta^{(a,b,c,d)}(x)$ of
(3.14) both obey (5.11) for $m=1,2,...,N$, so their ratio must be a constant
as it is invariant under the action of each $\Gamma_{i}$. This constant is
$1$, by noting that $q^{-\lambda\cdot\mu}P^{(a,b,c,d)}(x;s)\to 1$ and
$\Delta^{(a,b,c,d)}\to 1$ in the limit when all $x^{-{\alpha}_{i}}\to 0$. The
Theorem follows. ∎

Using the specializations of the parameters $(a,b,c,d)$ as in Table 1,

######  Corollary 5.4.

The functions $Q^{({\mathfrak{g}})}(s;x)$ and $P^{({\mathfrak{g}})}(x;s)$ are
related via

(5.12)
$Q^{({\mathfrak{g}})}(s;x)=\frac{P^{({\mathfrak{g}})}(x;s)}{\Delta^{({\mathfrak{g}})}(x)},$

with $\Delta^{({\mathfrak{g}})}$ as in (3.29).

It is now a simple exercise to relate the universal $(a,b,c,d)$ Koornwinder-
Pieri solution to the universal solution of the $(a^{*},b^{*},c^{*},d^{*})$
Koornwinder-Macdonald eigenvalue equation. As a result we have the following
duality relation between universal Pieri solutions.

######  Theorem 5.5.

We have the following duality formulas:

(5.13) $Q^{(a^{*},b^{*},c^{*},d^{*})}(x;s)=Q^{(a,b,c,d)}(s;x),$

and their ${\mathfrak{g}}$ specializations:

(5.14) $Q^{({\mathfrak{g}}^{*})}(x;s)=Q^{({\mathfrak{g}})}(s;x).$

###### Proof.

For conciseness, we omit the superscripts $(a,b,c,d)$ and use the superscript
$*$ to stand for $(a^{*},b^{*},c^{*},d^{*})$. Starting from the equation
$\hat{\mathcal{H}}_{1}(s)\,Q(s;x)=\hat{e}_{1}(x)\,Q(s;x)$, using (5.8), we
have

${\mathcal{D}}_{1}^{*}(s)\,\Delta^{*}(s)Q(s;x)=\sigma
t^{N-1}\,\hat{e}_{1}(s)\,\Delta^{*}(s)Q(s;x).$

Interchanging the variables $x\leftrightarrow s$, we find that
$\Delta^{*}(x)Q(x;s)$ is a solution to the $(a^{*},b^{*},c^{*},d^{*})$
eigenvalue equation. Moreover, using the normalization of $Q(x;s)$ with $s$
and $x$ interchanged, we have for small $\\{x^{-{\alpha}_{i}}\\}$:
$\Delta^{*}(x)Q(x;s)=q^{\lambda\cdot\mu}\big{(}1+O(\\{x^{-{\alpha}_{i}}\\})\big{)}$.
We conclude that $\Delta^{*}(x)Q(x;s)=P^{*}(x;s)$ by uniqueness of the
solution. The Theorem follows from $P^{*}(x;s)=\Delta^{*}(x)Q^{*}(s;x)$ by
Theorem 5.3 applied to $(a^{*},b^{*},c^{*},d^{*})$. ∎

This can alternatively be rephrased as duality between Koornwinder-Macdonald
eigenvalue universal solutions:

(5.15)
$\Delta^{(a^{*},b^{*},c^{*},d^{*})}(s)P^{(a,b,c,d)}(x;s)=\Delta^{(a,b,c,d)}(x)\,P^{(a^{*},b^{*},c^{*},d^{*})}(s;x),$

and their specializations:

(5.16)
$\Delta^{({\mathfrak{g}}^{*})}(s)\,P^{({\mathfrak{g}})}(x;s)=\Delta^{({\mathfrak{g}})}(x)\,P^{({\mathfrak{g}}^{*})}(s;x).$

Some of the above relations appear in different guises in the literature:
explicitly in the type A case [NS12] (see also Section 2.13), implicitly for
the other types [Che95b, Che97] where universal functions are obtained as
$x,s$-symmetric reproducing kernels. We now detail the explicit link between
the universal function duality relation (5.15) and the Koornwinder polynomial
duality (3.12).

######  Theorem 5.6.

The universal function duality relation (5.15) reduces to the Koornwinder
polynomial duality relation (3.12) upon specializing the variables
$x=q^{\mu}t^{\rho^{*}}$ and $s=q^{\lambda}t^{\rho}$ for $\lambda,\mu$ integer
partitions.

###### Proof.

Starting from the universal function $P^{(a,b,c,d)}(x;s)$, we use the
specialization $s=q^{\lambda}t^{\rho}$, leading to the Koornwinder polynomial
$P^{(a,b,c,d)}(x;q^{\lambda}t^{\rho})=t^{-\rho^{*}\cdot\lambda}\,P^{(a,b,c,d)}_{\lambda}(x)$.
Similarly, using the specialization $x=q^{\mu}t^{\rho^{*}}$ on the function
$P^{(a^{*},b^{*},c^{*},d^{*})}(s;x)$ leads to
$P^{(a^{*},b^{*},c^{*},d^{*})}(s,q^{\mu}t^{\rho^{*}})=t^{-\rho\cdot\mu}\,P^{(a^{*},b^{*},c^{*},d^{*})}_{\mu}(s)$.
The double specialization $s=q^{\lambda}t^{\rho},x=q^{\mu}t^{\rho^{*}}$ of
(5.15) results in

(5.17)
$t^{-\rho^{*}\cdot\lambda}\,\Delta^{(a^{*},b^{*},c^{*},d^{*})}(q^{\lambda}t^{\rho})\,P^{(a,b,c,d)}_{\lambda}(q^{\mu}t^{\rho^{*}})=t^{-\rho\cdot\mu}\,\Delta^{(a,b,c,d)}(q^{\mu}t^{\rho^{*}})\,P_{\mu}^{(a^{*},b^{*},c^{*},d^{*})}(q^{\lambda}t^{\rho}).$

The relation (3.12) follows from the following identity:

(5.18)
$\frac{\Delta^{(a^{*},b^{*},c^{*},d^{*})}(q^{\lambda}t^{\rho})}{\Delta^{(a,b,c,d)}(q^{\mu}t^{\rho^{*}})}=t^{\rho^{*}\cdot\lambda-\rho\cdot\mu}\,\frac{P_{\mu}^{(a^{*},b^{*},c^{*},d^{*})}(t^{\rho})}{P^{(a,b,c,d)}_{\lambda}(t^{\rho^{*}})},$

itself a consequence of (3.15), and of the identity
$\Delta^{(a^{*},b^{*},c^{*},d^{*})}(t^{\rho})=\Delta^{(a,b,c,d)}(t^{\rho^{*}})$.
The Theorem follows. ∎

Similarly, under the suitable specialization, the ${\mathfrak{g}}$-Macdonald
duality relation (5.16) reduces to the Macdonald polynomial duality (3.28) for
$x=q^{\mu}t^{\rho^{*}}$ and $s=q^{\lambda}t^{\rho}$, where $\lambda$ is any
${\mathfrak{g}}$-partition, and $\mu$ any ${\mathfrak{g}}^{*}$-partition.

####  5.2.2. Duality of universal solutions in the $q$-Whittaker limit

Universal solutions of the ${\mathfrak{g}}$-Macdonald eigenvector and
${\mathfrak{g}}$-Pieri equations simplify drastically in the $t\to\infty$
limit. They read:

(5.19)
$\Pi^{({\mathfrak{g}})}(x;{\Lambda}):=\lim_{t\to\infty}t^{\rho^{*}\cdot\lambda}\,P^{({\mathfrak{g}})}(x;{\Lambda}t^{\rho})=x^{\lambda}\sum_{\beta\in
Q_{+}(R)}c_{\beta}^{({\mathfrak{g}})}({\Lambda})\,x^{-\beta},$

and

(5.20) ${\bf
K}^{({\mathfrak{g}})}({\Lambda};x):=\lim_{t\to\infty}t^{\rho^{*}\cdot\lambda}\,Q^{({\mathfrak{g}})}({\Lambda}t^{\rho};x)=x^{\lambda}\sum_{\beta\in
Q_{+}(R^{*})}{\bar{c}}_{\beta}^{({\mathfrak{g}})}(x)\,{\Lambda}^{-\beta},$

with
$c_{0}^{({\mathfrak{g}})}({\Lambda})={\bar{c}}_{0}^{({\mathfrak{g}})}(x)=1$.
Note that the limit $t\to\infty$ has broken the previous symmetry
$x\leftrightarrow s$, as the $s$ variable itself contained a $t$-dependent
factor. However, Corollary 5.4 of the previous section turns into the
following.

######  Theorem 5.7.

The universal solutions $\Pi^{({\mathfrak{g}})}(x;{\Lambda})$ and ${\bf
K}^{({\mathfrak{g}})}({\Lambda};x)$ are related via:

(5.21)
$\Pi^{({\mathfrak{g}})}(x;{\Lambda})=\bar{\Delta}^{({\mathfrak{g}})}(x)\,{\bf
K}^{({\mathfrak{g}})}({\Lambda};x),\ \
\bar{\Delta}^{({\mathfrak{g}})}(x):=\lim_{t\to\infty}\Delta^{({\mathfrak{g}})}(x)=\prod_{\alpha\in\widehat{\mathcal{R}}_{++}({\mathfrak{g}})}(1-x^{-\alpha}).$

The eigenvalue and Pieri equations are:

(5.22) $\displaystyle
D_{a}^{({\mathfrak{g}})}(x)\,\Pi^{({\mathfrak{g}})}(x;{\Lambda})$
$\displaystyle=$
$\displaystyle{\Lambda}^{\omega_{a}^{*}}\,\Pi^{({\mathfrak{g}})}(x;{\Lambda}),$
(5.23) $\displaystyle H_{m}^{({\mathfrak{g}})}({\Lambda})\,{\bf
K}^{({\mathfrak{g}})}({\Lambda};x)$ $\displaystyle=$
$\displaystyle{\hat{e}}_{m}^{(R)}(x)\,{\bf
K}^{({\mathfrak{g}})}({\Lambda};x).$

Both equations for $a,m=1$ turn into simple triangular recursion relations for
the coefficients $c_{\beta}^{({\mathfrak{g}})}({\Lambda})$ and
${\bar{c}}_{\beta}^{({\mathfrak{g}})}(x)$ respectively.

###  5.3. Fourier transform and proof of the Macdonald- Q-system conjecture

Like in the case of type A, we may reformulate the Fourier transform (1.3) in
the $q$-Whittaker limit in terms of universal solutions via:

$f(x)\,\Pi^{({\mathfrak{g}})}(x;{\Lambda})={\bar{f}}({\Lambda})\,\Pi^{({\mathfrak{g}})}(x;{\Lambda})\quad\Leftrightarrow\quad
f(x)\,{\bf K}^{({\mathfrak{g}})}({\Lambda};x)={\bar{f}}({\Lambda})\,{\bf
K}^{({\mathfrak{g}})}({\Lambda};x).$

The main result of Section 4 is the relation (4.25) which expresses the fact
that $\bar{D}_{a;n}^{({\mathfrak{g}})}({\Lambda})$ is the Fourier transform of
$D_{a;n}^{({\mathfrak{g}})}(x)$. In terms of the universal Pieri solution, we
expect:

$D_{a;n}^{({\mathfrak{g}})}(x)\,{\bf
K}^{({\mathfrak{g}})}({\Lambda};x)=\bar{D}_{a;n}^{({\mathfrak{g}})}({\Lambda})\,{\bf
K}^{({\mathfrak{g}})}({\Lambda};x).$

As a consequence any relation satisfied by the $\bar{D}$’s is satisfied by the
$D$’s in the opposite direction, thus proving the Macdonald Q-system
conjecture. The proof of these identities is identical to that in Section 4,
and relies on the Fourier duality between $g^{({\mathfrak{g}})}({\Lambda})$
and $\gamma^{({\mathfrak{g}})}(x)$, whose adjoint action respectively
generates the discrete time translation in the ${\Lambda}$ and $x$ pictures:

$\gamma^{({\mathfrak{g}})}(x)\,{\bf
K}^{({\mathfrak{g}})}({\Lambda};x)=g^{({\mathfrak{g}})}({\Lambda})\,{\bf
K}^{({\mathfrak{g}})}({\Lambda};x).$

The proofs rely on a uniqueness argument which can be rephrased as follows.
The universal Pieri solution ${\bf K}^{({\mathfrak{g}})}({\Lambda};x)$ (5.20)
is fixed by the Pieri equation (5.23) for $m=1$, up to an overall
multiplicative function independent of ${\Lambda}$, and fixed by the leading
term normalization ${\bar{c}}_{0}^{({\mathfrak{g}})}(x)=1$. Any other series
solution of this Pieri equation is therefore proportional to ${\bf
K}^{({\mathfrak{g}})}({\Lambda};x)$, by a factor independent of ${\Lambda}$.

As we saw in Section 4.3.3, the case of odd times $a=N$ for
${\mathfrak{g}}=B_{N}^{(1)}$ required the use of Rains operators, and the
mapping to a companion theory $B_{N}^{(1)\,^{\prime}}$ with its own
$q$-Whittaker polynomials and quantum Q-system. Let us rephrase the action of
Rains operators at finite $t$ (3.8) in the $B_{N}^{(1)}$ specialization
$(a,b,c,d)=(t,-1,q^{\frac{1}{2}},-q^{\frac{1}{2}})$ of Table 1 in terms of
universal Macdonald functions. Let us denote for short $P(x;s)\equiv
P^{(B_{N}^{(1)})}(x;s)$ and $P^{\prime}(x;s)\equiv
P^{(B_{N}^{(1)\,^{\prime}})}(x;s)$ the respective universal Macdonald
solutions. As both $B_{N}^{(1)},B_{N}^{(1)\,^{\prime}}$ share the same
$\xi_{\mathfrak{g}}=\xi_{{\mathfrak{g}}^{\prime}}=\frac{1}{2}$, i.e.
$\sigma=\sigma^{\prime}=t^{\frac{1}{2}}$ and
$R({\mathfrak{g}})=R({\mathfrak{g}}^{\prime})$ while
$t^{\rho^{*}_{i}}=t^{N-i+1}$ and
$t^{(\rho^{\prime})^{*}_{i}}=q^{\frac{1}{2}}t^{N-i+1}$, we may rewrite (3.8)
as:

$\mathcal{R}_{N}^{(1,-1)}(x)\,P(x;s)=F(s)\,P^{\prime}(x;s),\qquad\mathcal{R}_{N}^{(t,-1)}(x)\,P^{\prime}(x;s)=F^{\prime}(s)\,P(x;s),$

with

$F(s)=\prod_{i=1}^{N}(1+t^{-\frac{1}{2}}s_{i}),\quad
F^{\prime}(s)=t^{N(N+1)/2}\prod_{i=1}^{N}(1+t^{-\frac{1}{2}}s_{i}^{-1}),$

where we have used the same notations as in the proof of Lemma A.10, with
$u=ab/\sigma=-t^{\frac{1}{2}}$, and taken into account the product formula
$q^{|\lambda|}=t^{-N^{2}/2}\,\prod_{i=1}^{N}s_{i}$. Taking the limit
$t\to\infty$ results in:

$R_{N}^{(0)}\,\Pi(x;{\Lambda})=({\Lambda}_{1}{\Lambda}_{2}\cdots{\Lambda}_{N})^{\frac{1}{2}}\,\left(1+\frac{1}{{\Lambda}_{N}}\right)\,\Pi^{\prime}(x;{\Lambda}),\
\
R_{N}^{(1)}\,\Pi^{\prime}(x;{\Lambda})=({\Lambda}_{1}{\Lambda}_{2}\cdots{\Lambda}_{N})^{\frac{1}{2}}\,\Pi(x;{\Lambda}).$

The steps of the proof can then be repeated identically, in particular
establishing that
$\gamma(x)\,\Pi^{\prime}(x;{\Lambda})=g^{\prime}({\Lambda})\,\Pi^{\prime}(x;{\Lambda})$,
and then
$D_{a;n}^{\prime}\,\Pi^{\prime}(x;{\Lambda})=\bar{D}_{a;n}^{\prime}\,\Pi^{\prime}(x;{\Lambda})$
for all $a\in[1,N]$ and $n\in{\mathbb{Z}}$, as well as finally
$D_{a;n}\,\Pi(x;{\Lambda})=\bar{D}_{a;n}\,\Pi(x;{\Lambda})$, from which the
main Theorems follow.

##  6\. Discussion

###  6.1. Companion quantum Q systems

Part of the proof of our main theorem, concerning the short label in type
$B_{N}^{(1)}$ (Section 4.3.3), revealed that acting with the Rains operators
(3.8) and their $\tau_{+}$-translates on the $q$-Whittaker functions gives
rise to a companion system $B_{N}^{(1)\,^{\prime}}$ 4.20 to the $B_{N}^{(1)}$
quantum Q-system. This new system has the same classical ($q\to 1$ limit) as
the $B_{N}^{(1)}$ quantum Q-system but corresponds to a different
specialization of the Koornwinder parameters $(a,b,c,d)$, and has different
Pieri operators and time translation operator (4.32).

One can obtain two further companion systems using the Rains operators,
illustrated in Figure 1, from the theories at specialization parameters
corresponding to ${\mathfrak{g}}=A_{2N-1}^{(2)},A_{2N}^{(2)}$. These also have
the same classical limit as the respective quantum Q-systems, and may be
considered as alternative quantizations. In type $A_{2N}^{(2)}$, which is not
a cluster algebra mutation in the first place, the companion system is, in
some sense, simpler and more natural from the quantization point of view, and
appears to be the one related to the graded tensor product character formulas
of KR-modules [Lin].

${B_{N}^{(1)}\\!\\!\\!}$${\\!\\!\\!\\!\\!\\!\\!\\!(t,-1,q^{\frac{1}{2}},-q^{\frac{1}{2}})}$${A_{2N}^{(2)}}$${\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(t,-1,t^{\frac{1}{2}}q^{\frac{1}{2}},-t^{\frac{1}{2}}q^{\frac{1}{2}})}$${B_{N}^{(1)\,^{\prime}}\\!\\!\\!}$${\\!\\!\\!\\!\\!\\!\\!\\!(1,-1,tq^{\frac{1}{2}},-q^{\frac{1}{2}})}$${A_{2N}^{(2)\,^{\prime}}}$${\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(q^{\frac{1}{2}}t,-q^{\frac{1}{2}},t^{\frac{1}{2}},-t^{\frac{1}{2}})}$${A_{2N-1}^{(2)}}$${\\!\\!\\!\\!\\!\\!\\!\\!(t^{\frac{1}{2}},-t^{\frac{1}{2}},q^{\frac{1}{2}},-q^{\frac{1}{2}})\quad}$${D_{N}^{(1)}\
\
}$${\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(1,-1,q^{\frac{1}{2}},-q^{\frac{1}{2}})}$${A_{2N-1}^{(2)\,^{\prime}}}$${\\!\\!\\!\\!\\!\\!\\!\\!(1,-1,t^{\frac{1}{2}}q^{\frac{1}{2}},-t^{\frac{1}{2}}q^{\frac{1}{2}})\quad}$${D_{N+1}^{(2)}\,\,}$${\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(t,-1,tq^{\frac{1}{2}},-q^{\frac{1}{2}})}$${C_{N}^{(1)}\
\
}$${\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(t^{\frac{1}{2}},-t^{\frac{1}{2}},t^{\frac{1}{2}}q^{\frac{1}{2}},-t^{\frac{1}{2}}q^{\frac{1}{2}})}$$\scriptstyle{\mathcal{R}^{(0)}_{N}}$$\scriptstyle{\mathcal{R}^{(2)}_{N}}$$\scriptstyle{\mathcal{R}^{(1)}_{N}}$$\scriptstyle{\mathcal{R}^{(1)}_{N}}$$\scriptstyle{\mathcal{R}^{(0)}_{N}}$$\scriptstyle{\mathcal{R}^{(0)}_{N}}$$\scriptstyle{\mathcal{R}^{(2)}_{N}}$$\scriptstyle{\mathcal{R}^{(1)}_{N}}$$\scriptstyle{\mathcal{R}^{(2)}_{N}}$
Figure 1. The nine families of Koornwinder-Macdonald operators/polynomials. We
have indicated the Rains operators that intertwine the various theories, and
the specializations of the Koornwinder parameters $(a,b,c,d)$.

Using the definition of the Rains operators (3.7), together with the
properties (3.8) of Section 3.1.3, from with the specializations of Table 1,
one obtains from the ${\mathfrak{g}}$-Macdonald polynomials
$P_{\lambda}^{({\mathfrak{g}})}(x)$ companion polynomials
$P_{\lambda}^{({\mathfrak{g}}^{\prime})}(x)$, using three different
specializations of the Rains operators,
${\mathcal{R}}_{N}^{(0)}={\mathcal{R}}_{N}^{(1,-1)}$,
${\mathcal{R}}_{N}^{(1)}={\mathcal{R}}_{N}^{(t,-1)}$ and
${\mathcal{R}}_{N}^{(2)}={\mathcal{R}}_{N}^{(t^{\frac{1}{2}},-t^{\frac{1}{2}})}$.
In the cases ${\mathfrak{g}}=D_{N}^{(1)},D_{N+1}^{(2)},C_{N}^{(1)}$, the Rains
operators leave the ${\mathfrak{g}}$-Macdonald polynomials invariant up to
some scalar, so that ${\mathfrak{g}}^{\prime}={\mathfrak{g}}$. In those cases,
the Rains operator is itself a Koornwinder-Macdonald operator, as shown in
Lemma A.13. Only the specializations
${\mathfrak{g}}=B_{N}^{(1)},A_{2N-1}^{(2)}$ and $A_{2N}^{(2)}$ are mapped to
new companion theories ${\mathfrak{g}}^{\prime}\neq{\mathfrak{g}}$.

In the $q$-Whittaker limit, the Rains operators tend to
$R_{N}^{(0)},R_{N}^{(1)}$ of (3.57-3.58) and to

$R^{(2)}_{N}=\lim_{t\to\infty}t^{-N(N+1)/2}{\mathcal{R}}_{N}^{(t^{\frac{1}{2}},-t^{\frac{1}{2}})}=\sum_{\epsilon_{1},\ldots,\epsilon_{N}=\pm
1}\prod_{i=1}^{N}\frac{x_{i}^{2\epsilon_{i}}}{x_{i}^{2\epsilon_{i}}-1}\prod_{1\leq
i<j\leq
N}\frac{x_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}}{x_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}-1}\prod_{i=1}^{N}\Gamma_{i}^{{\epsilon_{i}}/2}.$

These map the $q$-Whittaker polynomials
$\Pi_{\lambda}=\Pi_{\lambda}^{({\mathfrak{g}})}$ to a multiple of the monic
companions
$\Pi_{\lambda}^{\prime}=\Pi_{\lambda}^{({\mathfrak{g}}^{\prime})}=\lim_{t\to\infty}P_{\lambda}^{({\mathfrak{g}}^{\prime})}$.
More precisely (c.f. (4.30) for $B_{N}^{(1)}$), (3.8) implies the following
mapping of polynomials:

$\begin{array}[]{rll}A_{2N-1}^{(2)}:&R^{(0)}_{N}\,\Pi_{\lambda}=({\Lambda}_{1}{\Lambda}_{2}\cdots{\Lambda}_{N})^{\frac{1}{2}}\left(1+\frac{1}{{\Lambda}_{N}}\right)\Pi_{\lambda}^{\prime},&R^{(2)}_{N}\,\Pi_{\lambda}^{\prime}=({\Lambda}_{1}{\Lambda}_{2}\cdots{\Lambda}_{N})^{\frac{1}{2}}\Pi_{\lambda},\\\
A_{2N}^{(2)}:&R^{(2)}_{N}\,\Pi_{\lambda}=({\Lambda}_{1}{\Lambda}_{2}\cdots{\Lambda}_{N})^{\frac{1}{2}}\Pi_{\lambda}^{\prime},&R^{(1)}_{N}\,\Pi_{\lambda}^{\prime}=({\Lambda}_{1}{\Lambda}_{2}\cdots{\Lambda}_{N})^{\frac{1}{2}}\Pi_{\lambda}.\end{array}$

For the new companion cases
$A_{2N-1}^{(2)\,^{\prime}},A_{2N}^{(2)\,^{\prime}}$, we define companion
Koornwinder-Macdonald operators by specializing Definition A.8 for
$m=1,2,...,N$ at the appropriate values of the parameters as in Figure 1.
Their $q$-Whittaker limits are denoted by
$D_{m}^{\prime}=D_{m}^{({\mathfrak{g}}^{\prime})}$, and their
$\tau_{+}$-translates are defined for all $a\in[1,N]$ and $n\in{\mathbb{Z}}$
as follows:

$\displaystyle A_{2N-1}^{(2)\,^{\prime}}:\
D^{\prime}_{a;2n+i}=q^{-na}\,\gamma^{-n}\,D^{\prime}_{a;i}\,\gamma^{n},\quad(i=-1,0)$
$\displaystyle A_{2N}^{(2)\,^{\prime}}:\ \ \ \ \ \
D^{\prime}_{a;n}=q^{-na/2}\,\gamma^{-n}\,D^{\prime}_{a}\,\gamma^{n},$

where $D^{\prime}_{a;-1}$ is defined as in (3.50) with the suitable function
$\phi^{({\mathfrak{g}}^{\prime})}_{i,\epsilon}(x)$. Note that all labels of
$A_{2N-1}^{(2)\,^{\prime}}$ are short and all labels of
$A_{2N}^{(2)\,^{\prime}}$ are long, which is the reverse of the case for the
un-primed theories. Using the methods of this paper, it can be shown that
these new operators obey new quantum Q-systems, with the same $q$-commutation
relations and recursion relations as their non-primed companions, except for
the label $N$, where

$\begin{array}[]{rl}A_{2N-1}^{(2)\,^{\prime}}:&q^{N}\,Q_{N,2n+2}\,Q_{N,2n}=Q_{N,2n+1}^{2}-q^{-2n}\,Q_{N-1,2n+1}^{2},\\\
&q^{N}\,Q_{N,2n+1}\,Q_{N,2n-1}=(Q_{N,2n}-q^{-n}Q_{N-1,2n})(Q_{N,2n}+q^{1-n}Q_{N-1,2n}),\\\
A_{2N}^{(2)\,^{\prime}}:&q^{N}\,Q_{N,n+1}\,Q_{N,n-1}=Q_{N,n}^{2}-q^{-\frac{n}{2}}Q_{N,n}Q_{N-1,n}.\end{array}$

for all $n\in{\mathbb{Z}}$. To derive these results, we first define the
candidate Fourier transforms $\bar{D}^{\prime}_{a;n}$ via the opposite quantum
Q-systems and same initial data as the non-primed companions (that is, the
same values of $\omega_{a},\omega_{a}^{*}$), and compute the time translation
operators:

$\displaystyle A_{2N-1}^{(2)\,^{\prime}}:\ \ \
g^{\prime}=g_{T}g_{\Lambda}\frac{1}{({\Lambda}_{N}^{-2};q^{2})_{\infty}}g_{T}g_{\Lambda}\frac{1}{({\Lambda}_{N}^{-1};q)_{\infty}(-q{\Lambda}_{N}^{-1};q)_{\infty}},$
$\displaystyle A_{2N}^{(2)\,^{\prime}}:\ \ \ \ \
g^{\prime}=g_{T}g_{\Lambda}\,\frac{1}{({\Lambda}_{N}^{-1};q)_{\infty}},$

which commute, respectively, with the two Pieri operators or Toda Hamiltonians

$\displaystyle A_{2N-1}^{(2)\,^{\prime}}:\ \ \
H_{1}^{\prime}({\Lambda})=T_{1}+\sum_{i=2}^{N}{\scriptstyle\left(1-\frac{{\Lambda}_{i}}{{\Lambda}_{i-1}}\right)}T_{i}+\sum_{i=1}^{N-1}{\scriptstyle\left(1-\frac{{\Lambda}_{i+1}}{{\Lambda}_{i}}\right)}T_{i}^{-1}+{\scriptstyle\left(1-\frac{1}{{\Lambda}_{N}}\right)\left(1+\frac{q}{{\Lambda}_{N}}\right)}T_{N}^{-1},$
$\displaystyle A_{2N}^{(2)\,^{\prime}}:\ \ \ \ \
H_{1}^{\prime}({\Lambda})=T_{1}+\sum_{i=2}^{N}{\scriptstyle\left(1-\frac{{\Lambda}_{i}}{{\Lambda}_{i-1}}\right)}T_{i}+\sum_{i=1}^{N-1}{\scriptstyle\left(1-\frac{{\Lambda}_{i+1}}{{\Lambda}_{i}}\right)}T_{i}^{-1}+{\scriptstyle\left(1-\frac{1}{{\Lambda}_{N}}\right)}T_{N}^{-1}-{\scriptstyle\frac{q^{-\frac{1}{2}}}{{\Lambda}_{N}}}$

obtained from the $q$-Whittaker limit of the specialized operators of Theorem
(B.1). After proving that
$g^{\prime}\Pi_{\lambda}=\gamma^{t_{1}}\Pi_{\lambda}^{\prime}$ ($t_{1}=2$ for
$A_{2N-1}^{(2)\,^{\prime}}$, $t_{1}=1$ for $A_{2N}^{(2)\,^{\prime}}$), one
concludes that
$D_{a;n}\Pi_{\lambda}^{\prime}=\bar{D}_{a;n}^{\prime}\Pi_{\lambda}^{\prime}$
and the quantum Q-system relations for $D_{a;n}$ follow.

Finally, matching the specialization of the formula (3.14) for
$\Delta^{({\mathfrak{g}}^{\prime})}$ with the form of the product over affine
roots (3.29), we make the following identification of affine roots
corresponding to ${\mathfrak{g}}^{\prime}$:

$\displaystyle B_{N}^{(1)\,^{\prime}}:\ \ \
\widehat{R}_{++}=\scriptstyle{\left\\{(n-\frac{1}{2})\delta+e_{i}:\ 1\leq
i\leq N,\ n\geq 1\right\\}\sqcup\left\\{n\delta+(e_{i}\pm e_{j}):\ 1\leq
i<j\leq N,\ n\geq 1\right\\}}$ $\displaystyle A_{2N-1}^{(2)\,^{\prime}}:\
\widehat{R}_{++}=\scriptstyle{\left\\{(2n-1)\delta+2e_{i}:\ 1\leq i\leq N,\
n\geq 1\right\\}\sqcup\left\\{n\delta+(e_{i}\pm e_{j}):\ 1\leq i<j\leq N,\
n\geq 1\right\\}}$ $\displaystyle A_{2N}^{(2)\,^{\prime}}:\ \ \
\widehat{R}_{++}=\scriptstyle{\left\\{(n-\frac{1}{2})\delta+e_{i},\,2n\delta+2e_{i},:\
1\leq i\leq N,\ n\geq 1\right\\}\sqcup\left\\{n\delta+(e_{i}\pm e_{j}):\ 1\leq
i<j\leq N,\ n\geq 1\right\\}}.$

###  6.2. Universal solutions and Path models

In Section 5, we introduced universal solutions $P(x;s)$ and $Q(s;x)$ for the
various eigenvalue equations and Pieri rules, by solving an infinite
triangular linear system of equations for the expansion coefficients in each
case. Remarkably, the solutions can all be put in the form of path models,
such as that for the $q$-Whittaker vectors in the $A_{N-1}^{(1)}$ case
[DFKT17], see also [GT21]. As shown in Example 2.16, this results in an
interpretation of the solution of the triangular system as a partition
function for weighted paths on the relevant positive root cone $Q_{+}$ or
$Q_{+}^{*}$.

Let us start with the case of $Q(s;x)=q^{\lambda\cdot\mu}\sum_{\beta\in
Q_{+}^{*}}{\bar{c}}_{\beta}(x)\,s^{-\beta}$, $\bar{c}_{0}(x)=1$, the series
solution of the Pieri equation
${\mathcal{H}}_{1}(s)\,Q(s;x)=\hat{e}_{1}(x)\,Q(s;x)$ (either the Koornwinder
or ${\mathfrak{g}}$-Macdonald Pieri rule). The important fact is that
${\mathcal{H}}_{1}(s)$ is a $q$-difference operator in the variable $s$ of the
form ${\mathcal{H}}_{1}(s)=\sum_{i=1}^{N}\sum_{\epsilon=0,\pm
1}h_{i,\epsilon}(s)T_{i}^{\epsilon}$, where the coefficients $h_{i,\epsilon}$
are rational functions of the variables $\\{s^{-{\alpha}_{i}^{*}}\\}$, with
${\alpha}_{i}^{*}$ the simple roots of $R^{*}$. Factoring out the common
denominator in the coefficients, $h_{i,\epsilon}(s)=p_{i,\epsilon}(s)/q(s)$
with $p_{i,\epsilon}(s)$ and $q(s)$ some polynomials in
$\\{s^{-{\alpha}_{i}^{*}}\\}$, the Pieri equation is equivalent to
$\left(\hat{e}_{1}(x)\,q(s)-\sum_{i=1}^{N}\sum_{\epsilon=0,\pm
1}p_{i,\epsilon}(s)x_{i}^{\epsilon}T_{i}^{\epsilon}\right){q^{-\lambda\cdot\mu}Q(s;x)}=0.$
The difference operator can be written as $\sum_{i=1}^{N}\sum_{\epsilon=0,\pm
1}\sum_{{\alpha}\in
U^{*}}h_{i,\epsilon,{\alpha}}(x)s^{-{\alpha}}T_{i}^{\epsilon}$ for some
$x$-dependent coefficients $h_{i,\epsilon,{\alpha}}(x)$, where ${\alpha}$ is
summed over a finite subset $U^{*}$ of $Q_{+}^{*}$. Noting that
$T_{i}^{\epsilon}\,s^{-\beta}=q^{-\epsilon e_{i}\cdot\beta}s^{-\beta}$ and
that $s^{-{\alpha}}\,s^{-\beta}=s^{-({\alpha}+\beta)}$, and collecting the
coefficient of $s^{-\beta}$ in the Pieri equation, we obtain
$\sum_{i=1}^{N}\sum_{\epsilon=0,\pm 1}\sum_{{\alpha}\in
U^{*}}h_{i,\epsilon,{\alpha}}(x)\,q^{-\epsilon
e_{i}\cdot(\beta-{\alpha})}\,{\bar{c}}_{\beta-{\alpha}}(x)=0,$ or equivalently

$\left(\sum_{i,\epsilon}h_{i,\epsilon,0}(x)q^{-\epsilon
e_{i}\cdot\beta}\right){\bar{c}}_{\beta}(x)=-\sum_{i,\epsilon}\sum_{{\alpha}\in
U^{*}\setminus\\{0\\}}h_{i,\epsilon,{\alpha}}(x)q^{-\epsilon
e_{i}\cdot(\beta-{\alpha})}{\bar{c}}_{\beta-{\alpha}}(x).$

The factor on the left hand side is non-vanishing for generic $x$ and is
therefore invertible. The path model interpretation goes as follows. Given the
initial data $\bar{c}_{0}(x)=1$, the coefficient ${\bar{c}}_{\beta}(x)$ is the
partition function of paths from $0$ to $\beta$ in $Q^{*}_{+}$, consisting of
steps in the finite subset $U^{*}\setminus\\{0\\}$. Each path has a weight
equal to the product of vertex and edge weights along the path, defined
respectively as

$w_{\varphi}=\left(\sum_{i,\epsilon}h_{i,\epsilon,0}(x)q^{-\epsilon
e_{i}\cdot\varphi}\right)^{-1},\quad w_{0}=1,\quad
w_{\varphi-{\alpha},\varphi}=-\sum_{i,\epsilon}h_{i,\epsilon,{\alpha}}(x)q^{-\epsilon
e_{i}\cdot(\varphi-{\alpha})}$

for any $\varphi\in Q_{+}^{*}$, ${\alpha}\in U^{*}\setminus\\{0\\}$. The
weight of a path $p$ is a product over its vertex and edge sets $v(p),e(p)$:

$w(p)=\prod_{\varphi\in
v(p)}w_{\varphi}\times\prod_{(\varphi-{\alpha},\varphi)\in
e(p)}w_{\varphi-{\alpha},\varphi},$

and the coefficient $\bar{c}_{\beta}(x)$ is the partition function of all such
paths

${\bar{c}}_{\beta}(x)=\sum_{{\rm paths}\,p\ {\rm in}\ Q_{+}^{*}\atop{\rm
from}\ 0\to\beta}w(p).$

A similar argument leads to a formulation of the Macdonald eigenvalue equation
universal solution $P(x;s)$ in terms of a path model on $Q_{+}$.

The path model for the Pieri solution simplifies drastically in the
$q$-Whittaker limit as the Hamiltonians are directly polynomials of the
${\Lambda}^{-{\alpha}_{i}^{*}}$, giving rise to a small set of steps
$U^{*}\setminus\\{0\\}$. For instance, by inspection of (3.46-3.47), we find
that for all ${\mathfrak{g}}$,
$w_{\varphi}^{-1}=\sum_{i=1}^{N}\sum_{\epsilon=\pm
1}x_{i}^{\epsilon}(1-q^{-\epsilon\beta\cdot e_{i}})$.

######  Example 6.1.

In the case ${\mathfrak{g}}=A_{4}^{(2)}$, the $q$-Whittaker limit of the Pieri
equation is

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(T_{1}+\left(1-\frac{{\Lambda}_{2}}{{\Lambda}_{1}}\right)(T_{2}+T_{1}^{-1})+\left(1-\frac{1}{{\Lambda}_{2}}\right)T_{2}^{-1}-\frac{1}{{\Lambda}_{2}}-(x_{1}+x_{1}^{-1}+x_{2}+x_{2}^{-1})\right)$
$\displaystyle\times$ $\displaystyle
x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2}}\sum_{n,m\geq
0}{\tilde{c}}_{n,m}(x)\left(\frac{{\Lambda}_{2}}{{\Lambda}_{1}}\right)^{n}\left(\frac{1}{{\Lambda}_{2}}\right)^{m}$

expressed in terms of the variables
${\Lambda}^{-{\alpha}_{1}}=\frac{{\Lambda}_{2}}{{\Lambda}_{1}},{\Lambda}^{-{\alpha}_{2}}=\frac{1}{{\Lambda}_{2}}$,
${\alpha}_{i}$ the simple roots of $B_{2}$, and with the normalization
${\tilde{c}}_{0,0}(x)=1$. The is equivalent to the recursion relation

${\tilde{c}}_{n,m}(x)=\frac{q^{n-1}(q^{-m}x_{2}+x_{1}^{-1}){\tilde{c}}_{n-1,m}(x)+(q^{m-n-1}x_{2}^{-1}+1){\tilde{c}}_{n,m-1}(x)}{x_{1}(q^{-n}-1)+x_{1}^{-1}(q^{n}-1)+x_{2}(q^{n-m}-1)+x_{2}^{-1}(q^{m-n}-1)},$

which has the form
${\tilde{c}}_{n,m}(x)=a_{n,m}{\tilde{c}}_{n-1,m}(x)+b_{n,m}{\tilde{c}}_{n,m-1}(x)$,
the solution is

${\tilde{c}}_{n,m}(x)=\sum_{{\rm paths}\,p:(0,0)\to(n,m)}\prod_{{\rm
steps}\,s\in p}w_{s}\ ,$

where the is sum over all the paths on ${\mathbb{Z}}_{+}^{2}$ with steps
$(1,0)$ and $(0,1)$, from the origin to $(n,m)$, of the product of step
weights $w_{s}$ where $w_{s}=a_{i,j}$ for a horizontal step $(i-1,j)\to(i,j)$
and $w_{s}=b_{i,j}$ for a vertical step $(i,j-1)\to(i,j)$.

###  6.3. Universal solutions and $q$-Whittaker functions

We have used the terminology $q$-Whittaker functions for the $t\to\infty$
limits of Koornwinder-Macdonald polynomials. Strictly speaking,
$\Pi_{\lambda}^{({\mathfrak{g}})}(x)$ is a class-$1$ $q$-Whittaker function
for the quantum universal enveloping algebra of a simple Lie algebra only in
the cases where the Pieri operators are known to be $q$-deformed quantum Toda
Hamiltonians. These are related to the quantum Q-system of types
${\mathfrak{g}}=A_{N-1}^{(1)},D_{N}^{(1)},D_{N+1}^{(2)},A_{2N-1}^{(2)}$, whose
conserved quantities are the relativistic Toda Hamiltonians, of types
$A_{N-1},D_{N},B_{N},C_{N}$, respectively (see Remark 3.10). By analogy, we
call all the eigenfunctions $\Pi_{\lambda}^{({\mathfrak{g}})}(x)$
$q$-Whittaker functions for all ${\mathfrak{g}}$ in this paper. These share a
number of properties.

In the $q$-Whittaker limit, the universal series solutions of the
${\mathfrak{g}}$-Macdonald eigenvalue equations and the Pieri equations
(5.19-5.20) of Section 5, when specialized to $\lambda$ a
${\mathfrak{g}}$-partition, reduce respectively to the analogues of class-1
$q$-Whittaker functions
$\Pi_{\lambda}^{({\mathfrak{g}})}(x)=\Pi^{({\mathfrak{g}})}(x;q^{\lambda})$,
and the analogues of fundamental $q$-Whittaker functions
$\Psi_{\lambda}^{({\mathfrak{g}})}(x)=\mathrm{K}^{({\mathfrak{g}})}(x;q^{\lambda})$.
The former is a Weyl-symmetric polynomial in $x$, whereas the latter is a non-
symmetric series solution of the relativistic Toda equation with prescribed
leading term. It is associated with the highest weight Verma module $V_{\mu}$,
where $\mu$ is obtained from the variable $x=q^{\rho+\mu}$. In the theory of
Whittaker functions, the fundamental Whittaker function is convergent only in
a particular Weyl chamber. The class-1 function is a linear combination of
fundamental Whittaker functions, regular in all the Weyl chambers. It is
obtained via a symmetrization over the Weyl group action on the fundamental
solutions with suitable coefficients.

A subtlety arises in the study of convergence of the above series. In the
definition of $\bar{\Delta}^{({\mathfrak{g}})}(x)$ as an infinite product, it
is assumed that $|q|<1$ for convergence. On the other hand, the series
$\mathrm{K}^{({\mathfrak{g}})}(q^{\lambda};x)$ or
$\Psi_{\lambda}^{({\mathfrak{g}})}(x)$ is well-behaved for $|q|>1$. Obviously
$\Pi_{\lambda}^{({\mathfrak{g}})}(x)$, being a polynomial, makes sense for
both $|q|>1$ and $|q|<1$. Note that the infinite product

$\tilde{\Delta}^{({\mathfrak{g}})}=\frac{1}{\prod_{n=0}^{\infty}\prod_{{\alpha}\in
R_{+}}(1-q^{-n}x^{-{\alpha}})}$

is another solution of (2.39), convergent for $|q|>1$ instead of $|q|<1$. (To
see this, note that conjugating $\Gamma_{i}^{\pm 1}$ with
$\bar{\Delta}^{({\mathfrak{g}})}(x)$ or $\tilde{\Delta}^{({\mathfrak{g}})}(x)$
yields the same result.). The remarkable fact is that as a convergent series
for $|q|>1$,
$\tilde{\Delta}^{({\mathfrak{g}})}(x)\,\Psi_{\lambda}^{({\mathfrak{g}})}(x)$
is not equal to the polynomial class-$1$ $q$-Whittaker function
$\Pi_{\lambda}^{({\mathfrak{g}})}(x)$. We conjecture777Such a symmetrization
formula exists relating classical fundamental and class-$1$ Whittaker
functions. that for general ${\mathfrak{g}}$ it requires a symmetrization over
the Weyl group, valid for $|q|>1$:

(6.1) $\Pi_{\lambda}^{({\mathfrak{g}})}(x)=\sum_{w\in
W}\tilde{\Delta}^{({\mathfrak{g}})}(wx)\,\Psi_{\lambda}^{({\mathfrak{g}})}(wx).$

We end up with two characterizations of $\mathrm{K}({\Lambda};x)$: (1) as a
series with $|q|<1$ in the variables ${\Lambda}^{-{\alpha}_{i}}$, equal to
$\Pi^{({\mathfrak{g}})}(x;{\Lambda})/\bar{\Delta}^{({\mathfrak{g}})}(x)$ (by
Theorem 5.7) and (2) as a series with $|q|>1$ equal to
$\Psi_{\lambda}^{({\mathfrak{g}})}(x)$ when $\lambda$ is a
${\mathfrak{g}}$-partition. The polynomials
$\Pi_{\lambda}^{({\mathfrak{g}})}(x)$ are expressed in terms of both, but with
very different formulas for $|q|<1$ and $|q|>1$.

######  Example 6.2.

Let us illustrate the above in the simplest case of $A_{1}^{(1)}$. Denoting by
$u=x_{2}/x_{1}$, $v={\Lambda}_{2}/{\Lambda}_{1}$,

$\displaystyle p(u;v)$ $\displaystyle:=$ $\displaystyle
x_{1}^{-\lambda_{1}}x_{2}^{-\lambda_{2}}\,\Pi(x;{\Lambda})=\sum_{n=0}^{\infty}u^{n}\,\prod_{i=0}^{n-1}\frac{1-vq^{i}}{1-q^{-i-1}},$
$\displaystyle k(v;u)$ $\displaystyle:=$ $\displaystyle
x_{1}^{-\lambda_{1}}x_{2}^{-\lambda_{2}}\,\mathrm{K}({\Lambda};x)=\sum_{n=0}^{\infty}v^{n}\,\frac{q^{n^{2}}u^{n}}{\prod_{i=1}^{n}(1-q^{i})(1-uq^{i})}.$

With $\bar{\Delta}(u)=(qu;q)_{\infty}$ and
$\tilde{\Delta}(u)=1/(u;q^{-1})_{\infty}$,

$\displaystyle p(u;v)$ $\displaystyle=$
$\displaystyle\bar{\Delta}(u)\,k(v;u)\qquad(|q|<1),$
$\displaystyle\Pi_{\lambda_{1},\lambda_{2}}(x)$ $\displaystyle=$
$\displaystyle
x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2}}\,\tilde{\Delta}(u)\,k(q^{\lambda_{2}-\lambda_{1}};u)+x_{2}^{\lambda_{1}}x_{1}^{\lambda_{2}}\,\tilde{\Delta}(u^{-1})\,k(q^{\lambda_{2}-\lambda_{1}};u^{-1})\quad(|q|>1),$
$\displaystyle=$ $\displaystyle
x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2}}\,p(u;q^{\lambda_{2}-\lambda_{1}})\quad(|q|<1),$

where the second and third lines hold for any integers
$\lambda_{1}\geq\lambda_{2}\geq 0$.

###  6.4. Summary/Perspectives

We have proved the Macdonald operator-quantum Q-system conjectures, which
state that suitably defined, $\tau_{+}$-translated ${\mathfrak{g}}$-Macdonald
difference operators, in the $q$-Whittaker limit, obey the
${\mathfrak{g}}$-type quantum Q-system relations, and may as such be
considered as cluster variables in a suitable quantum cluster algebra in all
cases except $A_{2N}^{(2)}$. The proofs cover the cases
${\mathfrak{g}}=A_{N-1}^{(1)},D_{N}^{(1)},B_{N}^{(1)},C_{N}^{(1)},A_{2N-1}^{(2)},A_{2N}^{(2)},D_{N+1}^{(2)}$,
and rely strongly on the duality between the Koornwinder/Macdonald eigenvalue
equations and the associated Pieri rules.

We have proved that the conserved quantities of the ${\mathfrak{g}}$-quantum
Q-systems are the ${\mathfrak{g}}$-Pieri operators in the $q$-Whittaker limit,
which in a number of cases can themselves be identified with known
$q$-difference Toda Hamiltonians. Our construction provides explicit
expressions for all (higher) Hamiltonians as well. It would be interesting to
explore the combinatorial content of these, possibly in the language of
cluster integrable systems [GK13].

We have constructed time-translation operators $g({\Lambda})$ for all
${\mathfrak{g}}$ that commute with the corresponding $q$-difference Toda
Hamiltonians. More generally, in the spirit of [SS18], it would be desirable
to construct commuting Baxter $Q$-operators $Q(u;{\Lambda})$ that coincide
with $g({\Lambda})^{-1}$ at $u=1$, but are in general quantum-dilogarithmic
generating functions for the Hamiltonians. In all cases but $A_{2N}^{(2)}$,
such a construction should exist in terms of mutations of the corresponding
quantum cluster algebra.

As another by-product of our proofs, we have unearthed a remarkable structure
involving three more companion theories $B_{N}^{(1)\,^{\prime}}$,
$A_{2N-1}^{(2)\,^{\prime}}$ and $A_{2N}^{(2)\,^{\prime}}$ and their associated
quantum Q-systems. It would be interesting to further investigate their
quantum Laurent property, as well as their combinatorial content, in
particular the meaning of the associated fermionic sums which in the known
cases provide q-multiplicities of decompositions of KR modules onto
irreducibles (see [DFK14, DFK18]).

The quantum Q-system relates to the $q$-Whittaker limit of Macdonald-
Koornwinder theory, but it is natural to consider the extension of some of our
results to finite $t$. The time-shifted Macdonald operators
$\mathcal{D}^{({\mathfrak{g}})}_{a;n}(x;q,t)$ by iterated conjugation by
$\gamma(x)$ were considered in [DFK19] in type $A_{N-1}^{(1)}$, and are
generators of the spherical DAHA or, in the limit $N\to\infty$, the quantum
toroidal algebra of $\mathfrak{gl}_{1}$ or the elliptic Hall algebra. In the
Koornwinder we again have elements of the corresponding spherical DAHA but the
limit $N\to\infty$ is still to be understood.

Moreover, in type $A_{N-1}^{(1)}$ a $t$-deformed analogue of the time
translation operator $g({\Lambda})$ was defined in [LNS20] (see Proposition
4.2), and conjectured to act diagonally on the so-called Non-stationary
Ruijsenaars function, introduced in [Sh19] as the (conjectural) universal
series solution to the elliptic Ruijsenaars operator eigenvalue equation, also
related to the geometry of affine Laumon spaces. We expect this operator to
play the same role for the finite $t$ macdonald case as $g(\Lambda)$ for the
q-Whittaker limit $t\to\infty$, i.e. to be the (Macdonald) Fourier transform
of the Gaussian for finite $t$. We believe there should exist a cluster
algebra formulation of this operator. It would also be extremely interesting
to investigate this operator for general Koornwinder theory.

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

We combine several constructions [Mac01, vD95, Rai05] 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.

###  A.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
$R^{*}$ [Mac01].

Let $\\{e_{i}\\}_{1\leq i\leq N}$ 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 $v=\sum_{i}v_{i}e_{i}$.

$\begin{array}[]{|c|l|l|}\hline\cr\hbox{Algebra}&\hbox{Positive roots
$R_{+}$}&\hbox{fundamental weights $\omega_{a}$}\\\ \hline\cr B_{N}&e_{i},\
i\in[1,N];&\displaystyle{\omega_{i}=\sum_{k=1}^{i}e_{k},\ i\in[1,N-1]};\\\
&e_{i}\pm e_{j},\ 1\leq i<j\leq
N&\displaystyle{\omega_{N}=\frac{1}{2}\sum_{k=1}^{N}e_{k}}\\\ \hline\cr
C_{N}&e_{i}\pm e_{j},\ 1\leq i<j\leq N;\ 2e_{i},\
i\in[1,N]&\displaystyle{\omega_{i}=\sum_{k=1}^{i}e_{k},\ i\in[1,N]}\\\
\hline\cr D_{N}&e_{i}\pm e_{j},\ 1\leq i<j\leq
N&\displaystyle{\omega_{i}=\sum_{k=1}^{i}e_{k},\ i\in[1,N-2];}\\\
&&\displaystyle{\omega_{N}=\frac{1}{2}\sum_{k=1}^{N}e_{k},\
\omega_{N-1}=\omega_{N}-e_{N}}\\\ \hline\cr\end{array}$ Table 4. Positive
roots and fundamental weights of the finite dimensional algebras of types BCD.

There is a surjective map ${}_{*}:{R}\to{S}$,
${\alpha}\mapsto{\alpha}_{*}={\alpha}/u_{\alpha}\in S$, for some real
$u_{\alpha}$. In the case where $R\neq S$,
$u_{\alpha}=\frac{({\alpha},{\alpha})}{2}$, 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 define888We choose Macdonald’s parameters
$t_{\alpha}=t$ for all $\alpha$, independently of the length ${\alpha}$. This
allows us to obtain the dual $q$-Whittaker limit by simply taking
$t\to\infty$.

(A.1)
$\Phi_{\pi}:=\prod_{{\alpha}\in{R}_{+}\atop(\pi,{\alpha}_{*})=1}\frac{1-tx^{\alpha}}{1-x^{\alpha}},\qquad\Gamma_{\pi}=\prod_{i}\Gamma_{i}^{\pi_{i}}.$

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

(A.2) ${\mathcal{E}}_{\pi}f=\sum_{w\in
W}w\left(\Phi_{\pi}\Gamma_{\pi}f\right).$

We list below the explicit formulas for each case treated in [Mac01]. The
construction refers to the positive roots and fundamental weights for the
simple Lie groups of types $BCD$ in Table 4.

####  A.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

$\displaystyle\Phi_{\omega_{1}}$ $\displaystyle=$
$\displaystyle\prod_{j=2}^{N}\frac{1-tx_{1}x_{j}}{1-x_{1}x_{j}}\frac{tx_{1}-x_{j}}{x_{1}-x_{j}},\
\ \Phi_{\omega_{N-1}}=\prod_{1\leq i<j\leq
N-1}\frac{1-tx_{i}x_{j}}{1-x_{i}x_{j}}\prod_{i=1}^{N-1}\frac{tx_{i}-x_{N}}{x_{i}-x_{N}},$
$\displaystyle\Phi_{\omega_{N}}$ $\displaystyle=$ $\displaystyle\prod_{1\leq
i<j\leq N}\frac{1-tx_{i}x_{j}}{1-x_{i}x_{j}}.$

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

(A.3) $\displaystyle{\mathcal{E}}_{\omega_{1}}^{(D_{N}^{(1)})}$
$\displaystyle=$ $\displaystyle\sum_{\epsilon=\pm 1}\sum_{i=1}^{N}\prod_{j\neq
i}\frac{1-tx_{i}^{\epsilon}x_{j}}{1-x_{i}^{\epsilon}x_{j}}\frac{tx_{i}^{\epsilon}-x_{j}}{x_{i}^{\epsilon}-x_{j}}\Gamma_{i}^{\epsilon},$
(A.4) $\displaystyle{\mathcal{E}}_{\omega_{N-1}}^{(D_{N}^{(1)})}$
$\displaystyle=$ $\displaystyle\sum_{\epsilon_{1},...,\epsilon_{N}=\pm
1\atop\epsilon_{1}\epsilon_{2}\cdots\epsilon_{N}=-1}\prod_{1\leq i<j\leq
N}\frac{1-tx_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}}{1-x_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}}\prod_{i=1}^{N}\Gamma_{i}^{\frac{\epsilon_{i}}{2}},$
(A.5) $\displaystyle{\mathcal{E}}_{\omega_{N}}^{(D_{N}^{(1)})}$
$\displaystyle=$ $\displaystyle\sum_{\epsilon_{1},...,\epsilon_{N}=\pm
1\atop\epsilon_{1}\epsilon_{2}\cdots\epsilon_{N}=1}\prod_{1\leq i<j\leq
N}\frac{1-tx_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}}{1-x_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}}\prod_{i=1}^{N}\Gamma_{i}^{\frac{\epsilon_{i}}{2}}.$

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

####  A.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

$\Phi_{\omega_{1}}=\frac{1-tx_{1}}{1-x_{1}}\prod_{j=2}^{N}\frac{1-tx_{1}x_{j}}{1-x_{1}x_{j}}\frac{tx_{1}-x_{j}}{x_{1}-x_{j}}.$

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

${\mathcal{E}}_{\omega_{1}}^{(B_{N}^{(1)})}=\sum_{\epsilon=\pm
1}\sum_{i=1}^{N}\frac{1-tx_{i}^{\epsilon}}{1-x_{i}^{\epsilon}}\prod_{j\neq
i}\frac{1-tx_{i}^{\epsilon}x_{j}}{1-x_{i}^{\epsilon}x_{j}}\frac{tx_{i}^{\epsilon}-x_{j}}{x_{i}^{\epsilon}-x_{j}}\Gamma_{i}^{\epsilon}.$

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

####  A.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

$\Phi_{\omega_{N}}=\prod_{i=1}^{N}\frac{1-tx_{i}^{2}}{1-x_{i}^{2}}\prod_{1\leq
i<j\leq N}\frac{1-tx_{i}x_{j}}{1-x_{i}x_{j}}.$

The Weyl group is is the same as for type $B_{N}$, resulting in the difference
operator

(A.6)
${\mathcal{E}}_{\omega_{N}}^{(C_{N}^{(1)})}=\sum_{\epsilon_{1},...,\epsilon_{N}=\pm
1}\prod_{i=1}^{N}\frac{1-tx_{i}^{2\epsilon_{i}}}{1-x_{i}^{2\epsilon_{i}}}\prod_{1\leq
i<j\leq
N}\frac{1-tx_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}}{1-x_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}}\prod_{i=1}^{N}\Gamma_{i}^{\frac{\epsilon_{i}}{2}}.$

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

####  A.1.4. Macdonald operator for ${\mathfrak{g}}=A_{2N-1}^{(2)}$

Here, $(R,S)=(C_{N},B_{N})$. The map $*:R\to S$ 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

$\Phi_{\omega_{1}}=\frac{1-tx_{1}^{2}}{1-x_{1}^{2}}\prod_{j=2}^{N}\frac{1-tx_{1}x_{j}}{1-x_{1}x_{j}}\frac{tx_{1}-x_{j}}{x_{1}-x_{j}}.$

Summing over the Weyl group of type $C_{N}$,

${\mathcal{E}}_{\omega_{1}}^{(A_{2N-1}^{(2)})}=\sum_{\epsilon=\pm
1}\sum_{i=1}^{N}\frac{1-tx_{i}^{2\epsilon}}{1-x_{i}^{2\epsilon}}\prod_{j\neq
i}\frac{1-tx_{i}^{\epsilon}x_{j}}{1-x_{i}^{\epsilon}x_{j}}\frac{tx_{i}^{\epsilon}-x_{j}}{x_{i}^{\epsilon}-x_{j}}\Gamma_{i}^{\epsilon}.$

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

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

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.

####  A.1.5. Macdonald operator for ${\mathfrak{g}}=D_{N+1}^{(2)}$

Here, $(R,S)=(B_{N},C_{N})$. The map $*:R\to S$ 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

$\Phi_{\omega_{N}}=\prod_{i=1}^{N}\frac{1-tx_{i}}{1-x_{i}}\prod_{1\leq i<j\leq
N}\frac{1-tx_{i}x_{j}}{1-x_{i}x_{j}}.$

Summing over the Weyl group of type $C_{N}$ gives

(A.7)
${\mathcal{E}}_{\omega_{N}}^{(D_{N+1}^{(2)})}=\sum_{\epsilon_{1},...,\epsilon_{N}=\pm
1}\prod_{i=1}^{N}\frac{1-tx_{i}^{\epsilon_{i}}}{1-x_{i}^{\epsilon_{i}}}\prod_{1\leq
i<j\leq
N}\frac{1-tx_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}}{1-x_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}}\prod_{i=1}^{N}\Gamma_{i}^{\epsilon_{i}/2}.$

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

###  A.2. Higher order Koornwinder-Macdonald difference 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 [vD95]
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
[Rai05]. Upon specialization of the parameters (a,b,c,d), we combine this in
Sect. A.2.4, with Macdonald’s construction of Section A.1.

####  A.2.1. van Diejen’s higher order Koornwinder difference operators

######  Definition A.2.

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

(A.8) ${\mathcal{V}}_{m}^{(a,b,c,d)}:=\sum_{J\subset[1,N],\
|J|=m\atop\epsilon_{j}=\pm 1\ \forall\ j\in
J}\sum_{s=1}^{m}(-1)^{s-1}\sum_{\emptyset\subsetneq
J_{1}\subsetneq\cdots\subsetneq
J_{s}=J}\prod_{r=1}^{s}V_{\\{x\\},\\{\epsilon\\};J_{r}\setminus
J_{r-1};K_{r}}^{(a,b,c,d)}(\prod_{j\in J_{1}}\Gamma_{j}^{\epsilon_{j}}-1),$

where $J_{0}=\emptyset$, $K_{r}=[1,N]\setminus J_{r}$ and

(A.9) $\displaystyle V_{\\{x\\},\\{\epsilon\\};J;K}^{(a,b,c,d)}$
$\displaystyle:=$ $\displaystyle\prod_{i\in
J}\frac{(1-ax_{i}^{\epsilon_{i}})(1-bx_{i}^{\epsilon_{i}})(1-cx_{i}^{\epsilon_{i}})(1-dx_{i}^{\epsilon_{i}})}{(1-x_{i}^{2\epsilon_{i}})(1-qx_{i}^{2\epsilon_{i}})}$
$\displaystyle\times\prod_{{i<j}\atop{i,j\in
J}}\frac{1-tx_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}}{1-x_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}}\frac{1-qtx_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}}{1-qx_{i}^{\epsilon_{i}}x_{j}^{\epsilon_{j}}}\prod_{i\in
J\atop j\in
K}\frac{1-tx_{i}^{\epsilon_{i}}x_{j}}{1-x_{i}^{\epsilon_{i}}x_{j}}\frac{tx_{i}^{\epsilon_{i}}-x_{j}}{x_{i}^{\epsilon_{i}}-x_{j}},$

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 Equation
(3.2).

######  Example A.3.

Define

(A.10)
$A^{(a,b,c,d)}(x)=\frac{(1-ax)(1-bx)(1-cx)(1-dx)}{(1-x^{2})(1-qx^{2})}.$

Then

$\displaystyle{\mathcal{V}}_{2}^{(a,b,c,d)}$ $\displaystyle=$
$\displaystyle\sum_{1\leq i_{1}<i_{2}\leq N\atop\epsilon_{1},\epsilon_{2}=\pm
1}\prod_{\ell=1}^{2}\left(A^{(a,b,c,d)}(x_{i_{\ell}}^{\epsilon_{\ell}})\prod_{k\neq
i_{1},i_{2}}\frac{1-tx_{i_{\ell}}^{\epsilon_{\ell}}x_{k}}{1-x_{i_{\ell}}^{\epsilon_{\ell}}x_{k}}\frac{tx_{i_{\ell}}^{\epsilon_{\ell}}-x_{k}}{x_{i_{\ell}}^{\epsilon_{\ell}}-x_{k}}\right)$
$\displaystyle\times\left\\{\frac{1-tx_{i_{1}}^{\epsilon_{1}}x_{i_{2}}^{\epsilon_{2}}}{1-x_{i_{1}}^{\epsilon_{1}}x_{i_{2}}^{\epsilon_{2}}}\frac{1-qtx_{i_{1}}^{\epsilon_{1}}x_{i_{2}}^{\epsilon_{2}}}{1-qx_{i_{1}}^{\epsilon_{1}}x_{i_{2}}^{\epsilon_{2}}}(\Gamma_{i_{1}}^{\epsilon_{1}}\Gamma_{i_{2}}^{\epsilon_{2}}-1)-\frac{1-tx_{i_{1}}^{\epsilon_{1}}x_{i_{2}}}{1-x_{i_{1}}^{\epsilon_{1}}x_{i_{2}}}\frac{tx_{i_{1}}^{\epsilon_{1}}-x_{i_{2}}}{x_{i_{1}}^{\epsilon_{1}}-x_{i_{2}}}(\Gamma_{i_{1}}^{\epsilon_{1}}-1)\right.$
$\displaystyle\qquad\qquad\qquad\left.-\frac{1-tx_{i_{2}}^{\epsilon_{2}}x_{i_{1}}}{1-x_{i_{2}}^{\epsilon_{2}}x_{i_{1}}}\frac{tx_{i_{2}}^{\epsilon_{2}}-x_{i_{1}}}{x_{i_{2}}^{\epsilon_{2}}-x_{i_{1}}}(\Gamma_{i_{2}}^{\epsilon_{2}}-1)\right\\},$

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

$\displaystyle u^{(k)}$ $\displaystyle:=$ $\displaystyle\\{\sigma
t^{k-i}\\}_{1\leq i\leq k},$ $\displaystyle v$ $\displaystyle:=$
$\displaystyle\\{s_{i}+s_{i}^{-1}\\}_{1\leq i\leq N},$ $\displaystyle w_{(m)}$
$\displaystyle:=$ $\displaystyle\\{\sigma
t^{N-i}+\sigma^{-1}t^{-(N-i)}\\}_{m\leq i\leq N},$

with $s_{i}=\sigma t^{N-i}q^{\lambda_{i}}$ as usual, and the functions

(A.11) $\displaystyle d_{m}^{(k)}$ $\displaystyle:=$
$\displaystyle\sigma^{m}\,t^{m(k-\frac{m+1}{2})}\,{\hat{e}}_{m}(u^{(k)}),$
(A.12) $\displaystyle d_{\lambda;m}$ $\displaystyle:=$
$\displaystyle\sigma^{m}\,t^{m(N-\frac{m+1}{2})}\,{\hat{e}}_{m}(s),$ (A.13)
$\displaystyle f_{\lambda;m}$ $\displaystyle:=$
$\displaystyle\sigma^{m}\,t^{m(N-\frac{m+1}{2})}\,\sum_{j=0}^{m}(-1)^{j}\,e_{m-j}(v)\,h_{j}(w_{(m)}).$

The spectral theorem for van Diejen operators is

######  Theorem A.5.

[vD95] The (monic) symmetric Koornwinder polynomials
$P_{\lambda}^{(a,b,c,d)}(x)$ satisfy

(A.14)
${\mathcal{V}}_{m}^{(a,b,c,d)}\,P_{\lambda}^{(a,b,c,d)}(x)=f_{\lambda;m}\,P_{\lambda}^{(a,b,c,d)}(x).$

####  A.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 $\beta\leq N$, 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 $r\geq 0$ and $n\geq 1$,

(A.15)
$\theta_{r,n}(u):=\sum_{\ell=0}^{r}\,(-1)^{\ell}e_{r-\ell}(u^{[n-\ell]})h_{\ell}(u^{[n-\ell+1]})=\delta_{r,0}.$

###### 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 $\\{1,...,p^{\prime}\\}$ with
$j^{\prime}$ particles, denoted by $B$, is a sequence of $j^{\prime}$ integers
in $[1,p^{\prime}]$ 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
$j^{\prime}$ bosons on $[1,p^{\prime}]$ is
$h_{j^{\prime}}(-u^{[p^{\prime}]})=(-1)^{j^{\prime}}h_{j^{\prime}}(u^{[p^{\prime}]})$.

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

$\mathcal{S}_{n,r}=\left\\{(F,B)|\,t_{F}:=\max(F)\leq n-|B|,t_{B}:=\max(B)\leq
n-|B|+1,|F|+|B|=r\right\\}$

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:

$Z_{n,r}:=\sum_{(F,B)\in\mathcal{S}_{n,r}}w_{F,B}=\delta_{r,0}.$

To prove this, for any $r\geq 1$ 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:

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

Figure 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 $t_{F}^{\prime}=t_{B}$. Case (b) has
$t_{F}=6\geq t_{B}=5$, hence we move the fermionic particle to a bosonic one
at position $t_{B^{\prime}}=t_{F}$.

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 $\Phi(F,B)=(F^{\prime},B^{\prime})$, where

  * (a)

If $t_{B}>t_{F}$: $F^{\prime}=F\cup t_{B}$, $B^{\prime}=B\setminus t_{B}$.
Since $|B^{\prime}|=|B|-1$, $t_{F^{\prime}}=t_{B}\leq n-|B|+1=n-|B^{\prime}|$
and $t_{B^{\prime}}\leq t_{B}\leq n-|B|+1=n-|B^{\prime}|<n-|B^{\prime}|+1$.
Therefore, $(F^{\prime},B^{\prime})\in\mathcal{S}_{n,r}$, while
$w_{F^{\prime},B^{\prime}}=-w_{F,B}$.

  * (b)

If $t_{B}\leq t_{F}$: $F^{\prime}=F\setminus t_{F}$ and $B^{\prime}=B\cup
t_{F}$. Then $t_{F^{\prime}}\leq t_{F}-1\leq n-|B|-1=n-|B^{\prime}|$ and
$t_{B^{\prime}}=t_{F}\leq n-|B|=n-|B^{\prime}|+1$, so that
$(F^{\prime},B^{\prime})\in\mathcal{S}_{n,r}$, and
$w_{F^{\prime},B^{\prime}}=-w_{F,B}$.

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. ∎

######  Lemma A.7.

There is an identity on symmetric functions:

(A.16)
${\hat{e}}_{m}(x)=\sum_{j=0}^{m}{\hat{e}}_{m-j}({y}^{[N-j]})\sum_{k=0}^{j}(-1)^{k}e_{j-k}({\tilde{x}})h_{k}({\tilde{y}}^{[N-j+1]}).$

###### Proof.

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

(A.17)
${\hat{E}}(z;x)|_{z^{m}}=\sum_{j=0}^{m}{\hat{E}}(z;{y}^{[N-j]})|_{z^{m-j}}F(z;{\tilde{x}},{\tilde{y}}^{[N-j+1]})|_{z^{j}},$

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

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

One can further decompose

$\prod_{i=1}^{N-j}(1+z{\tilde{y}}_{i}+z^{2})=(1+z^{2})^{N-j}{\hat{E}}(\frac{z}{1+z^{2}};\tilde{y})=\sum_{k=0}^{N-j}z^{k}\,(1+z^{2})^{N-j-k}\,e_{k}(\tilde{y}^{[N-j]}),$

and

$\frac{\prod_{i=1}^{N}(1+z{\tilde{x}}_{i})}{\prod_{i=1}^{N-j+1}(1+z{\tilde{y}}_{i})}=\prod_{i=1}^{N}(1+z{\tilde{x}}_{i})\sum_{\ell\geq
0}(-1)^{\ell}z^{\ell}h_{\ell}(\tilde{y}^{[N-j+1]}).$

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

$\displaystyle\sum_{j=0}^{m}\sum_{k,\ell}(1+z^{2})^{N-j-k}\big{|}_{z^{m-j-k}}\,\prod_{i=1}^{N}(1+z{\tilde{x}}_{i})|_{z^{j-\ell}}\,(-1)^{\ell}e_{k}(\tilde{y}^{[N-j]})h_{\ell}({\tilde{y}}^{[N-j+1]})$
$\displaystyle=$ $\displaystyle\sum_{r=0}^{m}\sum_{k,\ell\geq 0\atop
k+\ell=r}\,\sum_{j=\ell}^{m-k}(1+z^{2})^{N-j-k}\big{|}_{z^{m-j-k}}\,\prod_{i=1}^{N}(1+z{\tilde{x}}_{i})|_{z^{j-\ell}}\,(-1)^{\ell}e_{k}(\tilde{y}^{[N-j]})h_{\ell}(\tilde{y}^{[N-j+1]}),$
$\displaystyle=$
$\displaystyle\sum_{r=0}^{m}\sum_{j=0}^{m-r}(1+z^{2})^{N-j-r}\big{|}_{z^{m-j-r}}\,\prod_{i=1}^{N}(1+z{\tilde{x}}_{i})|_{z^{j}}\,\sum_{\ell=0}^{r}\,(-1)^{\ell}e_{r-\ell}(\tilde{y}^{[N-j-\ell]})h_{\ell}(\tilde{y}^{[N-j-\ell+1]}),$

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

$\sum_{j=0}^{m}(1+z^{2})^{N-j}\big{|}_{z^{m-j}}\,\prod_{i=1}^{N}(1+z{\tilde{x}}_{i})|_{z^{j}}=\prod_{i=1}^{N}(1+z{\tilde{x}}_{i}+z^{2})\big{|}_{z^{m}}={\hat{e}}_{m}(x),$

which implies (A.17). ∎

######  Definition A.8.

We define the Koornwinder-Macdonald difference operators as

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

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):

(A.19)
${\mathcal{D}}_{m}^{(a,b,c,d)}(x;q,t)\,P_{\lambda}^{(a,b,c,d)}(x)=d_{\lambda;m}\,P_{\lambda}^{(a,b,c,d)}(x).$

###### Proof.

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

(A.20) $d_{\lambda;m}=\sum_{j=0}^{m}d_{j}^{(N-m+j)}\,f_{\lambda;m-j}.$

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. ∎

####  A.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:

(A.21)
$\widehat{\mathcal{D}}_{N}^{(a,b,c,d)}(x;q,t)={\mathcal{D}}_{N}^{(a,b,c,d)}(x;q,t)+\sum_{m=1}^{N}(-1)^{m}\left(a^{m}b^{m}+\frac{c^{m}d^{m}}{q^{m}}\right)\,t^{m(m-1)/2}\,{\mathcal{D}}_{N-m}^{(a,b,c,d)}(x;q,t).$

###### Proof.

This is a consequence of the relation between the eigenvalues:

(A.22)
$\hat{d}_{\lambda;N}^{(a,b,c,d)}=d_{\lambda,N}+\sum_{m=1}^{N}(-1)^{m}\left(a^{m}b^{m}+\frac{c^{m}d^{m}}{q^{m}}\right)\,t^{m(m-1)/2}\,d_{\lambda;N-m}.$

Using the notation $\sigma=\sqrt{abcd/q}$, $u=ab/\sigma$, $u^{-1}=cd/q\sigma$
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 $\hat{d}_{\lambda;N}$, Equation (A.22) is

$\displaystyle
q^{-|\lambda|}\prod_{i=1}^{N}(1-us_{i})(1-u^{-1}s_{i})=\sigma^{N}\,t^{N(N-1)/2}\left\\{\hat{e}_{N}(s)+\sum_{m=1}^{N}(-1)^{m}(u^{m}+u^{-m})\hat{e}_{N-m}(s)\right\\}$
$\displaystyle\ \
=\frac{(-1)^{N}\sigma^{N}\,t^{N(N-1)/2}}{u^{N}}\sum_{m=0}^{2N}(-1)^{m}u^{m}\hat{e}_{m}(s)=\frac{(-1)^{N}\sigma^{N}\,t^{N(N-1)/2}}{u^{N}}\prod_{i=1}^{N}(1-us_{i})(1-us_{i}^{-1}),$

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

####  A.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

(A.23) $\displaystyle{\mathcal{D}}_{1}^{(a,b,c,d)}(x;q,t)$ $\displaystyle=$
$\displaystyle{\mathcal{V}}_{1}^{(a,b,c,d)}(x;q,t)+\frac{1-t^{N}}{1-t}\left(1+\frac{abcd}{q}t^{N-1}\right),$
$\displaystyle=$
$\displaystyle{\mathcal{K}}_{1}^{(a,b,c,d)}(x;q,t)+\frac{1-t^{N}}{1-t}\left(1+\frac{abcd}{q}t^{N-1}\right).$

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

######  Lemma A.11.

The first order Koornwinder-Macdonald operator can be written as

$\qquad{\mathcal{D}}_{1}^{(a,b,c,d)}(x;q,t)=\varphi^{(a,b,c,d)}(x)+\sum_{i=1}^{N}\sum_{\epsilon=\pm
1}\Phi_{i,\epsilon}^{(a,b,c,d)}(x)\Gamma_{i}^{\epsilon},$

where

(A.24) $\displaystyle\varphi^{(a,b,c,d)}(x)$ $\displaystyle=$
$\displaystyle\sum_{\epsilon=\pm
1}\frac{\big{(}1-\frac{\epsilon}{q^{1/2}}a\big{)}\big{(}1-\frac{\epsilon}{q^{1/2}}b\big{)}\big{(}1-\frac{\epsilon}{q^{1/2}}c\big{)}\big{(}1-\frac{\epsilon}{q^{1/2}}d\big{)}}{2(1-t)(1-q^{-1}t)}$
$\displaystyle\qquad\qquad\qquad\times\left\\{\prod_{i=1}^{N}\frac{1-\frac{\epsilon\,t}{q^{1/2}}x_{i}}{1-\frac{\epsilon}{q^{1/2}}x_{i}}\,\frac{1-\frac{\epsilon\,t}{q^{1/2}}x_{i}^{-1}}{1-\frac{\epsilon}{q^{1/2}}x_{i}^{-1}}-t^{N}\right\\}.$

###### Proof.

We have:

(A.25)
$\varphi^{(a,b,c,d)}(x)=\frac{1-t^{N}}{1-t}\left(1+\frac{abcd}{q}t^{N-1}\right)-\sum_{i=1}^{N}\sum_{\epsilon=\pm
1}\Phi_{i,\epsilon}^{(a,b,c,d)}(x).$

Using the simple fraction decomposition of the function

(A.26)
$\theta(z):=\frac{\big{(}1-\frac{z}{a}\big{)}\big{(}1-\frac{z}{b}\big{)}\big{(}1-\frac{z}{c}\big{)}\big{(}1-\frac{z}{d}\big{)}}{\big{(}1-\frac{z^{2}}{q}\big{)}\big{(}1-\frac{z^{2}}{t}\big{)}}\,\prod_{i=1}^{N}\frac{1-\frac{z}{t}x_{i}}{1-zx_{i}}\,\frac{1-\frac{z}{t}x_{i}^{-1}}{1-zx_{i}^{-1}}.$

We find

$\theta(z)=\frac{q}{abcd\,t^{2N-1}}+\sum_{\epsilon=\pm
1}\left\\{\sum_{i=1}^{N}\frac{A_{i,\epsilon}}{1-zx_{i}^{\epsilon}}+\frac{B_{\epsilon}}{1-\epsilon\frac{z}{q^{1/2}}}+\frac{C_{\epsilon}}{1-\epsilon\frac{z}{t^{1/2}}}\right\\},$

where

$\displaystyle A_{i,\epsilon}$ $\displaystyle=$
$\displaystyle\frac{1-t}{t^{2N}}\,\frac{\big{(}1-\frac{x_{i}^{-\epsilon}}{a}\big{)}\big{(}1-\frac{x_{i}^{-\epsilon}}{b}\big{)}\big{(}1-\frac{x_{i}^{-\epsilon}}{c}\big{)}\big{(}1-\frac{x_{i}^{-\epsilon}}{d}\big{)}}{\big{(}1-\frac{x_{i}^{-2\epsilon}}{q}\big{)}\big{(}1-\frac{x_{i}^{-2\epsilon}}{t}\big{)}}\,\prod_{j\neq
i}\frac{tx_{i}^{\epsilon}-x_{j}}{x_{i}^{\epsilon}-x_{j}}\,\frac{tx_{i}^{\epsilon}x_{j}-1}{x_{i}^{\epsilon}x_{j}-1},$
$\displaystyle=$
$\displaystyle\frac{q(1-t)}{abcd\,t^{2N-1}}\,\Phi_{i,\epsilon}^{(a,b,c,d)}(x),$

and

$\displaystyle B_{\epsilon}$ $\displaystyle=$
$\displaystyle\frac{\big{(}1-\epsilon\frac{a}{q^{1/2}}\big{)}\big{(}1-\epsilon\frac{b}{q^{1/2}}\big{)}\big{(}1-\epsilon\frac{c}{q^{1/2}}\big{)}\big{(}1-\epsilon\frac{d}{q^{1/2}}\big{)}}{2\,abcd\,t^{2N-1}\,\big{(}1-\frac{t}{q}\big{)}}\prod_{i=1}^{N}\frac{1-\frac{\epsilon\,t}{q^{1/2}}x_{i}}{1-\frac{\epsilon}{q^{1/2}}x_{i}}\,\frac{1-\frac{\epsilon\,t}{q^{1/2}}x_{i}^{-1}}{1-\frac{\epsilon}{q^{1/2}}x_{i}^{-1}},$
$\displaystyle C_{\epsilon}$ $\displaystyle=$ $\displaystyle
q\frac{\big{(}1-\epsilon\frac{a}{t^{1/2}}\big{)}\big{(}1-\epsilon\frac{b}{t^{1/2}}\big{)}\big{(}1-\epsilon\frac{c}{t^{1/2}}\big{)}\big{(}1-\epsilon\frac{d}{t^{1/2}}\big{)}}{2\,abcd\,t^{N-1}\,\big{(}1-\frac{q}{t}\big{)}}.$

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

(A.27) $\sum_{\epsilon=\pm
1}\left\\{\sum_{i=1}^{N}\Phi_{i,\epsilon}^{(a,b,c,d)}(x)+\frac{abcd\,t^{2N-1}}{q(1-t)}\big{(}B_{\epsilon}+C_{\epsilon}\big{)}\right\\}=\frac{\frac{abcd}{q}\,t^{2N-1}-1}{1-t}.$

Using

$\displaystyle\sum_{\epsilon=\pm
1}\left\\{\frac{t}{2}\big{(}1-\epsilon\frac{a}{t^{1/2}}\big{)}\big{(}1-\epsilon\frac{b}{t^{1/2}}\big{)}\big{(}1-\epsilon\frac{c}{t^{1/2}}\big{)}\big{(}1-\epsilon\frac{d}{t^{1/2}}\big{)}\right.$
$\displaystyle\left.\qquad\qquad-\frac{q}{2}\big{(}1-\epsilon\frac{a}{q^{1/2}}\big{)}\big{(}1-\epsilon\frac{b}{q^{1/2}}\big{)}\big{(}1-\epsilon\frac{c}{q^{1/2}}\big{)}\big{(}1-\epsilon\frac{d}{q^{1/2}}\big{)}\right\\}=\left(1-\frac{q}{t}\right)\left(t-\frac{abcd}{q}\right),$

we have:

$\displaystyle\sum_{\epsilon=\pm
1}\frac{abcd\,t^{2N-1}}{q(1-t)}\,C_{\epsilon}=\frac{t^{N}}{2(1-t)\big{(}1-\frac{q}{t}\big{)}}\sum_{\epsilon=\pm
1}\big{(}1-\epsilon\frac{a}{t^{1/2}}\big{)}\big{(}1-\epsilon\frac{b}{t^{1/2}}\big{)}\big{(}1-\epsilon\frac{c}{t^{1/2}}\big{)}\big{(}1-\epsilon\frac{d}{t^{1/2}}\big{)},$
$\displaystyle=\frac{t^{N}-\frac{abcd}{q}t^{N-1}}{1-t}-\frac{t^{N}}{2(1-t)\big{(}1-\frac{t}{q}\big{)}}\sum_{\epsilon=\pm
1}\big{(}1-\epsilon\frac{a}{q^{1/2}}\big{)}\big{(}1-\epsilon\frac{b}{q^{1/2}}\big{)}\big{(}1-\epsilon\frac{c}{q^{1/2}}\big{)}\big{(}1-\epsilon\frac{d}{q^{1/2}}\big{)}.$

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). ∎

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:

(A.28)
${\mathcal{D}}_{1}^{({\mathfrak{g}})}(x;q,t)=\sum_{i=1}^{N}\sum_{\epsilon=\pm
1}\Phi_{i,\epsilon}^{({\mathfrak{g}})}(x)\Gamma_{i}^{\epsilon}={\mathcal{E}}_{1}^{({\mathfrak{g}})}(x;q,t).$

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

(A.29)
$\begin{array}[]{ll}D_{N}^{(1)}:&{\mathcal{E}}_{\omega_{N}}^{(D_{N}^{(1)})}(x;q,t)+{\mathcal{E}}_{\omega_{N-1}}^{(D_{N}^{(1)})}(x;q,t)={\mathcal{R}}_{N}^{(1,-1)}(x;q,t),\\\
&\widehat{\mathcal{D}}_{N}^{(1,-1,q^{1/2},-q^{1/2})}=\left({\mathcal{R}}_{N}^{(1,-1)}\right)^{2};\\\
C_{N}^{(1)}:&{\mathcal{E}}_{\omega_{N}}^{(C_{N}^{(1)})}(x;q,t)={\mathcal{R}}_{N}^{(t^{1/2},-t^{1/2})}(x;q,t),\\\
&\widehat{\mathcal{D}}_{N}^{(t^{1/2},-t^{1/2},t^{1/2}q^{1/2},-t^{1/2}q^{1/2})}=\left({\mathcal{R}}_{N}^{(t^{1/2},-t^{1/2})}\right)^{2};\\\
D_{N+1}^{(2)}:&{\mathcal{E}}_{\omega_{N}}^{(D_{N+1}^{(2)})}(x;q,t)={\mathcal{R}}_{N}^{(t,-1)}(x;q,t),\\\
&\widehat{\mathcal{D}}_{N}^{(t,-1,tq^{1/2},-q^{1/2})}=\left({\mathcal{R}}_{N}^{(t,-1)}\right)^{2}.\end{array}$

###  A.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.

####  A.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 Section A.1. In addition, there are non-linear
relations999These relations are proved using the same argument as in the proof
of Theorem A.9, using identities between eigenvalues. Some are a direct
consequence of the identifications with Rains operators from Theorem A.13.:

$\displaystyle{\mathcal{E}}_{\omega_{N}}^{(D_{N}^{(1)})}\,{\mathcal{E}}_{\omega_{N-1}}^{(D_{N}^{(1)})}$
$\displaystyle=$
$\displaystyle\sum_{{\alpha}=0}^{\lfloor\frac{N-1}{2}\rfloor}t^{{\alpha}(2{\alpha}+1)}\,{\mathcal{D}}^{(a,b,c,d)}_{N-2{\alpha}-1},$
$\displaystyle({\mathcal{E}}_{\omega_{N}}^{(D_{N}^{(1)})})^{2}+({\mathcal{E}}_{\omega_{N-1}}^{(D_{N}^{(1)})})^{2}$
$\displaystyle=$
$\displaystyle{\mathcal{D}}_{N}^{(a,b,c,d)}+2\sum_{{\alpha}=0}^{\lfloor\frac{N}{2}\rfloor}t^{{\alpha}(2{\alpha}-1)}\,{\mathcal{D}}^{(a,b,c,d)}_{N-2{\alpha}},$

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

$\left({\mathcal{E}}_{\omega_{N}}^{({\mathfrak{g}})}\right)^{2}={\mathcal{D}}_{N}^{(a,b,c,d)}+2\sum_{{\alpha}=1}^{N}t^{\frac{{\alpha}({\alpha}+1)}{2}}\,{\mathcal{D}}^{(a,b,c,d)}_{N-{\alpha}}.$

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.

####  A.3.2. The eigenvalues of ${\mathfrak{g}}$-Macdonald operators

######  Definition A.14.

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

(A.30)
$d_{\lambda,m}^{({\mathfrak{g}})}:=d_{\lambda,m}=t^{m(N+\xi_{\mathfrak{g}}-\frac{m+1}{2})}\,\hat{e}_{m}(s),\quad
m\in[1,N_{\mathfrak{g}}],$

where $s=t^{\rho^{({\mathfrak{g}})}}q^{\lambda}$, i.e.
$s_{i}=q^{\lambda_{i}}t^{N-i+\xi_{\mathfrak{g}}}$. Otherwise, define

(A.31)
$\begin{array}[]{ll}{\mathfrak{g}}=D_{N}^{(1)}:&d_{\lambda;\beta}^{(D_{N}^{(1)})}=t^{\frac{N(N-1)}{4}}\,{\hat{e}}_{\beta}^{(D_{N})}(s),\qquad\beta=N-1,N,\\\
{\mathfrak{g}}=C_{N}^{(1)},D_{N+1}^{(2)}:&d_{\lambda;N}^{({\mathfrak{g}})}=t^{\frac{N(N+1)}{4}}\,{\hat{e}}_{N}^{(B_{N})}(s),\end{array}$

where

$\displaystyle{\hat{e}}_{\beta}^{(D_{N})}(x)$ $\displaystyle:=$
$\displaystyle\sum_{\epsilon_{1},...,\epsilon_{N}=\pm
1\atop\prod\epsilon_{i}=2(\beta-N)+1}\prod_{i=1}^{N}x_{i}^{\epsilon_{i}/2}\qquad(\beta=N-1,N),$
(A.32) $\displaystyle{\hat{e}}_{N}^{(B_{N})}(x)$ $\displaystyle:=$
$\displaystyle\prod_{i=1}^{N}\frac{1+x_{i}}{\sqrt{x_{i}}}={\hat{e}}_{N-1}^{(D_{N})}(x)+{\hat{e}}_{N}^{(D_{N})}(x).$

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

(A.33)
${\mathcal{D}}^{({\mathfrak{g}})}_{m}(x)\,P_{\lambda}^{({\mathfrak{g}})}=d_{\lambda;m}^{({\mathfrak{g}})}\,P_{\lambda}^{({\mathfrak{g}})},\qquad
m\in[1,N],$

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

(A.34)
$\theta_{m}^{({\mathfrak{g}})}=t^{m(N+\xi_{\mathfrak{g}}-\frac{m+1}{2})},\qquad
m\leq N_{\mathfrak{g}},$ (A.35)
$\begin{array}[]{c}\theta_{\beta}^{(D_{N}^{(1)})}=t^{\frac{N(N-1)}{4}},\qquad\beta=N-1,N,\\\
\theta_{N}^{({\mathfrak{g}})}=t^{\frac{N(N+1)}{4}},\qquad{\mathfrak{g}}=C_{N}^{(1)},D_{N+1}^{(2)}.\end{array}$

####  A.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 Section A.1, the eigenvalue equation is

${\mathcal{D}}_{m}^{({\mathfrak{g}})}(x)\,P_{\lambda}^{({\mathfrak{g}})}(x)=\theta_{m}^{({\mathfrak{g}})}\,s_{\omega_{m}}^{(R^{*})}(s)\,P_{\lambda}^{({\mathfrak{g}})}(x),$

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 $e_{m}^{(R)}$:

$\displaystyle D_{N}:\quad s_{\omega_{m}}^{(D_{N})}$ $\displaystyle=$
$\displaystyle\hat{e}_{m}^{(D_{N})},\qquad m\in[1,N],$ $\displaystyle
B_{N}:\quad s_{\omega_{m}}^{(C_{N})}$ $\displaystyle=$
$\displaystyle\hat{e}_{m}-\hat{e}_{m-2},\qquad m\in[1,N],$ $\displaystyle
C_{N}:\quad s_{\omega_{m}}^{(B_{N})}$ $\displaystyle=$
$\displaystyle\hat{e}_{m}+\hat{e}_{m-1}\qquad m\in[1,N-1],$ $\displaystyle
s_{\omega_{N}}^{(B_{N})}$ $\displaystyle=$
$\displaystyle\hat{e}_{N}^{(B_{N})}.$

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

$\displaystyle{\mathfrak{g}}=D_{N}^{(1)}:\quad\tilde{\mathcal{D}}^{({\mathfrak{g}})}_{m}$
$\displaystyle:=$
$\displaystyle{\mathcal{D}}^{({\mathfrak{g}})}_{m}\qquad(m=1,2,...,N),$
$\displaystyle{\mathfrak{g}}=B_{N}^{(1)},A_{2N-1}^{(2)},A_{2N}^{(2)}:\quad\tilde{\mathcal{D}}^{({\mathfrak{g}})}_{m}$
$\displaystyle:=$
$\displaystyle{\mathcal{D}}^{({\mathfrak{g}})}_{m}-(1-\delta_{m,1})\frac{\theta_{m}^{({\mathfrak{g}})}}{\theta_{m-2}^{({\mathfrak{g}})}}\,{\mathcal{D}}^{({\mathfrak{g}})}_{m-2}\quad(m=1,2,...,N),$
$\displaystyle{\mathfrak{g}}=C_{N}^{(1)},D_{N+1}^{(2)}:\quad\tilde{\mathcal{D}}^{({\mathfrak{g}})}_{m}$
$\displaystyle:=$
$\displaystyle{\mathcal{D}}^{({\mathfrak{g}})}_{m}+\frac{\theta_{m}^{({\mathfrak{g}})}}{\theta_{m-1}^{({\mathfrak{g}})}}\,{\mathcal{D}}^{({\mathfrak{g}})}_{m-1}\quad(m=1,2,...,N-1),$
(A.36) $\displaystyle\tilde{\mathcal{D}}^{({\mathfrak{g}})}_{N}$
$\displaystyle:=$ $\displaystyle{\mathcal{D}}^{({\mathfrak{g}})}_{N},$

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 ${\mathfrak{g}}$,

(A.37)
$\tilde{\mathcal{D}}^{({\mathfrak{g}})}_{m}\,P_{\lambda}(x)=\theta_{m}^{({\mathfrak{g}})}\,s_{\omega_{m}}^{(R^{*})}(s)\,P_{\lambda}(x),\qquad
s=q^{\lambda}\,t^{\rho^{({\mathfrak{g}})}}.$

###  A.4. $q$-Whittaker limit of the ${\mathfrak{g}}$-Macdonald operators

The $q$-Whittaker limit corresponds to sending $t\to\infty$. 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):

$m\xi_{\mathfrak{g}}+m(m-1)+2m(N-m)=m(2N-m-1+\xi_{\mathfrak{g}}).$

Therefore,