# Supersymmetric Shimura operators and interpolation polynomials

Siddhartha SAHI  Department of Mathematics, Rutgers University
Hill Center for the Mathematical Sciences
110 Frelinghuysen Rd.
Piscataway, NJ 08854-8019<EMAIL_ADDRESS>and  Songhao ZHU  Department
of Mathematics, Rutgers University
Hill Center for the Mathematical Sciences
110 Frelinghuysen Rd.
Piscataway, NJ 08854-8019<EMAIL_ADDRESS>
###### Abstract.

The Shimura operators are a certain distinguished basis for invariant
differential operators on a Hermitian symmetric space. Answering a question of
Shimura, Sahi–Zhang showed that the Harish-Chandra images of these operators
are specializations of certain $BC$-symmetric interpolation polynomials that
were defined by Okounkov.

We consider the analogs of Shimura operators for the Hermitian symmetric
superpair $(\mathfrak{g},\mathfrak{k})$ where
$\mathfrak{g}=\mathfrak{gl}(2p|2q)$ and
$\mathfrak{k}=\mathfrak{gl}(p|q)\oplus\mathfrak{gl}(p|q)$ and we prove their
Harish-Chandra images are specializations of certain $BC$-supersymmetric
interpolation polynomials introduced by Sergeev–Veselov.

##  1\. Introduction

The Harish-Chandra homomorphism for a symmetric space gives an explicit
isomorphism of the algebra of invariant differential operators with a certain
polynomial algebra, and knowing the Harish-Chandra image of an operator allows
one to determine its spectrum. In [Shi90] Shimura introduced a basis for the
algebra of invariant differential operators on a Hermitian symmetric space,
and formulated the problem of determining their eigenvalues. Shimura’s problem
was solved in [SZ19] where it was shown that the Harish-Chandra images of
Shimura operators are specializations of certain interpolation polynomials of
Type BC introduced by Okounkov [Oko98].

Our main result, Theorem A below, solves the analogous problem for the
Hermitian symmetric superpair $(\mathfrak{g},\mathfrak{k})$ with
$\mathfrak{g}=\mathfrak{gl}(2p|2q)$ and
$\mathfrak{k}=\mathfrak{gl}(p|q)\oplus\mathfrak{gl}(p|q)$.

Let $\mathfrak{U}$ be the universal enveloping algebra of $\mathfrak{g}$, then
the algebra of invariant differential operators is the quotient of
$\mathfrak{k}$-invariants
$\mathfrak{D}:=\mathfrak{U}^{\mathfrak{k}}/(\mathfrak{U}\mathfrak{k})^{\mathfrak{k}}$.
In Section 3.3 we describe a basis $\mathscr{D}_{\mu}$ of $\mathfrak{D}$
indexed by the set $\mathscr{H}(p,q)$ of $(p,q)$-hook partitions, which are
partitions $\mu$ that satisfy $\mu_{p+1}\leq q$. These $\mathscr{D}_{\mu}$ are
the super analogs of Shimura operators, and their definition involves the
Cartan decomposition
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}=\mathfrak{k}\oplus\mathfrak{p}^{+}\oplus\mathfrak{p}^{-}$,
and the multiplicity-free $\mathfrak{k}$-decompositions of
$\mathfrak{S}(\mathfrak{p}^{+})$ and $\mathfrak{S}(\mathfrak{p}^{-})$, which
have $\mathfrak{k}$-summands $W_{\mu}$ and $W_{\mu}^{*}$, naturally indexed by
$\mu\in\mathscr{H}(p,q)$ [CW01].

The Harish-Chandra homomorphism (Section 2) is an algebra map
$\gamma^{\mathtt{0}}:\mathfrak{D}\to\mathfrak{P}(\mathfrak{a}^{*})$, where
$\mathfrak{a}$ is an even Cartan subspace of $\mathfrak{p}$ and
$\mathfrak{P}(\mathfrak{a}^{*})$ is the algebra of polynomials on
$\mathfrak{a}^{*}$. For our pair $(\mathfrak{g},\mathfrak{k})$ the image of
$\gamma^{\mathtt{0}}$ can be identified with the ring
$\mathscr{S}^{\mathtt{0}}_{p,q}$ of even supersymmetric polynomials in $p+q$
variables. In Section 6 we introduce a basis
$\\{J_{\mu}:\mu\in\mathscr{H}(p,q)\\}$ of $\mathscr{S}^{\mathtt{0}}_{p,q}$.
These $J_{\mu}$ are suitable specializations of the supersymmetric
interpolation polynomials of Type BC introduced by Sergeev–Veselov [SV09],
which are characterized up to multiple by certain vanishing properties.

######  Theorem A.

The Harish-Chandra image of $\mathscr{D}_{\mu}$ is $k_{\mu}J_{\mu}$ where
$k_{\mu}$ is explicitly given in Eq. (55).

We will deduce this from two other results. First, let $\mathfrak{Z}$ be the
center of $\mathfrak{U}$, then we have a natural map
$\pi:\mathfrak{Z}\hookrightarrow\mathfrak{U}^{\mathfrak{k}}\twoheadrightarrow\mathfrak{D}$,
and we prove the following result.

######  Theorem B.

The map $\pi$ is surjective. In particular, there exist
$Z_{\mu}\in\mathfrak{Z}$ such that $\pi(Z_{\mu})=\mathscr{D}_{\mu}$.

For $\lambda\in\mathscr{H}(p,q)$ let $I_{\lambda}$ be the generalized Verma
module for $\mathfrak{g}$ obtained by parabolic induction from
$\mathfrak{q}=\mathfrak{k}\oplus\mathfrak{p}^{+}$ of the
$\mathfrak{k}$-summand $W_{\lambda}$ of $\mathfrak{S}(\mathfrak{p}^{+})$
(Section 5).

######  Theorem C.

The central element $Z_{\mu}$ acts on $I_{\lambda}$ by $0$ unless
$\lambda_{i}\geq\mu_{i}$ for all $i$.

The special case $p=q=1$ of Theorem A was proved previously in [Zhu22], where
it was also shown that the general case would follow if one knew that certain
irreducible finite dimensional $\mathfrak{g}$-modules are
$\mathfrak{k}$-spherical [Zhu22, Conjecture 1, Theorem A]. For ordinary Lie
algebras, the Cartan–Helgason Theorem [Hel00] provides necessary and
sufficient conditions for $\mathfrak{k}$-sphericity. However its precise
analog for Lie superalgebras remains open and seems quite hard. Therefore in
this paper we give a different argument for Theorem A, which does not rely on
the $\mathfrak{k}$-sphericity conjecture.

The Shimura problem discussed in this paper is closely related to the Capelli
problem studied in [KS93, Sah94] for usual Lie algebras, in [SS16, ASS18], and
[SSS20] for Lie superalgebra, and more recently [LSS23, LSS22b, LSS22a] for
quantum groups. Thus it is possible that the results of this paper can be
generalized to many of the Hermitian symmetric superpairs constructed using
Jordan superalgebras, e.g. in [SSS20, Theorem 1.4], and also to the quantum
setting, and we hope to address this in future work.

We now briefly discuss the proofs of these results.

The proof of Theorem C is a generalization of an analogous argument in [SZ19]
and relies on two key properties of $I_{\lambda}$, namely that it is
$\mathfrak{k}$-spherical and has a natural grading by the center of
$\mathfrak{k}$.

For Theorem B let $\mathfrak{h}=\mathfrak{a}\oplus\mathfrak{t}_{+}$ be a
Cartan subalgebra of $\mathfrak{g}$ containing $\mathfrak{a}$, let
$\gamma:\mathfrak{Z}\to\mathfrak{P}(\mathfrak{h}^{*})$ be the usual Harish-
Chandra homomorphism [Mus12], and consider the diagram

(1)
${\mathfrak{Z}}$${\mathfrak{D}}$${\mathfrak{P}(\mathfrak{h}^{*})}$${\mathfrak{P}(\mathfrak{a}^{*})}$$\scriptstyle{\pi}$$\scriptstyle{\gamma}$$\scriptstyle{\gamma^{\mathtt{0}}}$$\scriptstyle{\operatorname{\mathtt{Res}}}$

where $\operatorname{\mathtt{Res}}$ is the restriction map induced by the
decomposition $\mathfrak{h}=\mathfrak{a}\oplus\mathfrak{t}_{+}$. We first show
that the diagram commutes and then we prove that the image of $\gamma$
surjects onto the image of $\gamma^{\mathtt{0}}$ using an explicit description
of generators of $\mathscr{S}^{\mathtt{0}}_{p,q}$ from [Ste85]. This implies
Theorem B.

For Theorem A the main point is to show that
$\gamma^{\mathtt{0}}(\mathscr{D}_{\mu})$ satisfies the vanishing properties
which characterize $J_{\mu}$. The action of $Z_{\mu}$ on $I_{\lambda}$ can be
expressed in terms of its Harish-Chandra image and thus Theorem C, implies
certain vanishing properties for $\gamma(Z_{\mu})$. Now we use Diagram 1 to
deduce the necessary vanishing properties for
$\gamma^{\mathtt{0}}(\mathscr{D}_{\mu})$. This last step requires an
additional result that we deduce from [AS15]: consider the set of $\lambda$
for which the irreducible quotient $V_{\lambda}$ of $I_{\lambda}$ is
$\mathfrak{k}$-spherical, and let $S$ be the set of highest
$\mathfrak{a}$-weights of these $V_{\lambda}$, then $S$ is a Zariski dense
subset of $\mathfrak{a}^{*}$.

The structure of the paper is as follows. In Section 2, we review some known
results related to the pair
$(\mathfrak{gl}(2p|2q),\mathfrak{gl}(p|q)\oplus\mathfrak{gl}(p|q))$. In
particular, we discuss the Iwasawa decomposition, the two Harish-Chandra
homomorphisms, the Cheng–Wang decomposition of
$\mathfrak{S}(\mathfrak{p}^{\pm})$, and Alldridge’s necessary conditions for
$\mathfrak{k}$-sphericity of finite dimensional irreducible
$\mathfrak{g}$-modules. We also present the Sergeev–Veselov supersymmetric
interpolation polynomials of Type BC. In Section 3, we define the
supersymmetric Shimura operators. In Section 4, we describe the image of
$\gamma^{\mathtt{0}}$ as the ring of even supersymmetric polynomials. We prove
the surjectivity of $\operatorname{\mathtt{Res}}$ in Subsection 4.1
(Proposition 4.1) and Theorem B in Subsection 4.2. In Section 5 we discuss the
generalized Verma modules and prove Theorem C, which provides the necessary
representation theoretic machinery for the vanishing properties. Finally in
Section 6, we reformulate the Type BC interpolation polynomials in our
context. Then we prove Theorem A using the results obtained in previous
sections.

##  2\. Preliminaries

Throughout the paper, we assume that $\mathbb{N}$, the set of natural numbers,
includes 0. For a finite dimensional vector space $V$, we identify the
polynomial algebra on $V$, $\mathfrak{P}(V)$, with the symmetric algebra
$\mathfrak{S}(V^{*})$ on the dual of $V$. We denote the $\mathbb{C}$-vector
superspace of superdimension $m|n$ by $\mathbb{C}^{m|n}$ where the even
subspace is $\mathbb{C}^{m}$ and the odd subspace is $\mathbb{C}^{n}$. A
$\mathbb{Z}_{2}$-graded algebra is called a superalgebra. For any superalgebra
$A$, we denote its even subspace as $A_{\overline{0}}$ and the odd subspace as
$A_{\overline{1}}$. For $x\in A_{i}$, we write $|x|=i$ for its parity $i$. The
space of linear endomorphisms $\operatorname{End}(\mathbb{C}^{m|n})$ is
naturally $\mathbb{Z}_{2}$-graded and identified with the space of
$(m+n)\times(m+n)$ matrices. We denote the Lie superalgebra structure defined
on $\operatorname{End}(\mathbb{C}^{m|n})$ as $\mathfrak{gl}(m|n)$. In this
subsection, we denote the standard diagonal Cartan subalgebra of
$\mathfrak{gl}(m|n)$ as $\mathfrak{t}$, that is,
$\mathfrak{t}=\operatorname{Span}\\{E_{i,i}:1\leq i\leq m+n\\}$ where
$E_{i,j}$ denotes the matrix with 1 in the $(i,j)$-th entry and 0 elsewhere.
We also let $\epsilon_{i}$ and $\delta_{j}$ be the coordinate functions of
$E_{i,i}$ for $1\leq i\leq m$ and of $E_{m+j,m+j}$ for $1\leq j\leq n$
respectively.

###  2.1. Symmetric superpairs

In this subsection, we let $(\mathfrak{g},\mathfrak{k})$ be a symmetric
superpair. Specifically, the Cartan decomposition
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ is given by an involution
$\theta$ where $\mathfrak{k}$ is the fixed point subalgebra and $\mathfrak{p}$
is the $(-1)$-eigenspace of $\theta$. We assume the chosen nondegenerate
invariant form $b$ on $\mathfrak{g}$ is $\theta$-invariant. Let
$\mathfrak{a}\subseteq\mathfrak{p}_{\overline{0}}$ be a maximal toral
subalgebra. Let $\Sigma:=\Sigma(\mathfrak{g},\mathfrak{a})$ be the restricted
root system of $\mathfrak{g}$ with respect to $\mathfrak{a}$, and $\Sigma^{+}$
be a positive system. We denote the form on $\mathfrak{a}^{*}$ by
$(\cdot,\cdot)$ induced from $b$. As in Subsection 2.2, we say a restricted
root $\alpha$ is anisotropic if $(\alpha,\alpha)\neq 0$, and isotropic
otherwise. We write $\mathfrak{g}_{\alpha}$ for the root space of $\alpha$,
and let

$\Sigma_{\overline{0}}:=\\{\alpha\in\Sigma:\mathfrak{g}_{\alpha}\cap\mathfrak{g}_{\overline{0}}\neq
0\\},\quad\Sigma_{\overline{1}}:=\\{\alpha\in\Sigma:\mathfrak{g}_{\alpha}\cap\mathfrak{g}_{\overline{1}}\neq
0\\}$

be the sets of even and odd restricted roots respectively. Set
$\Sigma^{+}_{\overline{0}}=\Sigma^{+}\cap\Sigma_{\overline{0}}$. If
$\alpha\in\Sigma$, but $\alpha/2\notin\Sigma$, we say $\alpha$ is indivisible.
We define the multiplicity of $\alpha$ as
$m_{\alpha}:=\dim(\mathfrak{g}_{\alpha})_{\overline{0}}-\dim(\mathfrak{g}_{\alpha})_{\overline{1}}$.
For a positive system $\Sigma^{+}$, we define the nilpotent subalgebra for
$\Sigma^{+}$ as
$\mathfrak{n}:=\bigoplus_{\alpha\in\Sigma^{+}}\mathfrak{g}_{\alpha}$. We
denote the restricted Weyl vector
$\frac{1}{2}\sum_{\alpha\in\Sigma^{+}}m_{\alpha}\alpha$ for $\Sigma^{+}$ as
$\rho$.

We assume

(2) $\mathfrak{g}=\mathfrak{n}\oplus\mathfrak{a}\oplus\mathfrak{k},$

which is called an Iwasawa decomposition for the pair
$(\mathfrak{g},\mathfrak{k})$. For an in-depth discussion and recent
developments on Iwasawa decomposition, see [She22].

The Poincaré–Birkhoff–Witt theorem applied to Eq. (2) yields the following
identity:

$\mathfrak{U}=\left(\mathfrak{U}\mathfrak{k}+\mathfrak{n}\mathfrak{U}\right)\oplus\mathfrak{S}(\mathfrak{a}).$

Let $\pi_{\rho}$ be the respective projection onto
$\mathfrak{S}(\mathfrak{a})\cong\mathfrak{P}(\mathfrak{a}^{*})$, and define
${\Gamma}_{\rho}(D)(\lambda):=\pi_{\rho}(D)(\lambda+\rho)$ on
$\mathfrak{U}^{\mathfrak{k}}$. The map ${\Gamma}_{\rho}$ is called the Harish-
Chandra homomorphism associated with $(\mathfrak{g},\mathfrak{k})$ (with
respect to the choice $\Sigma^{+}\subseteq\Sigma$). The quotient isomorphism

$\gamma^{\mathtt{0}}_{\rho}:\mathfrak{U}^{\mathfrak{k}}/\ker{\Gamma}_{\rho}\rightarrow\operatorname{Im}{\Gamma}_{\rho}$

is called the Harish-Chandra isomorphism associated with
$(\mathfrak{g},\mathfrak{k})$. Note
$\operatorname{Im}\gamma^{\mathtt{0}}_{\rho}=\operatorname{Im}{\Gamma}_{\rho}$.

Next, we introduce Alldridge’s result on spherical representations. Suppose
$\mathfrak{h}\supseteq\mathfrak{a}$ is a $\theta$-invariant Cartan subalgebra
extended from $\mathfrak{a}$. If $\mathfrak{b}\subseteq\mathfrak{g}$ is a
Borel subalgebra containing $\mathfrak{h}$, then we denote the irreducible
representation of highest weight $\lambda\in\mathfrak{h}^{*}$ with respect to
$\mathfrak{b}$ as $V(\lambda,\mathfrak{b})$. A $\mathfrak{g}$-module $V$ is
said to be $\mathfrak{k}$-spherical if $V^{\mathfrak{k}}:=\\{v\in
V:X.v=0\textup{ for all }X\in\mathfrak{k}\\}$ is non-zero. A non-zero vector
in $V^{\mathfrak{k}}$ is called a $\mathfrak{k}$-spherical vector. When the
context is clear, we simply say spherical instead of $\mathfrak{k}$-spherical.
Define
$\lambda_{\alpha}:=\dfrac{(\lambda|_{\mathfrak{a}},\alpha)}{(\alpha,\alpha)}$
for anisotropic $\alpha\in\Sigma$. We say $\lambda$ is high enough if

  1. (1)

$(\lambda,\beta)>0$ for any isotropic root $\beta\in\Sigma^{+}$,

  2. (2)

$\lambda_{\alpha}+m_{\alpha}+2m_{2\alpha}>0$ and
$\lambda_{\alpha}+m_{\alpha}+m_{2\alpha}+1>0$ for any odd anisotropic
indivisible root $\alpha$.

Then by [AS15, Theorem 2.3, Corollary 2.7], we have the following assertion.

######  Theorem 2.1.

If $\lambda_{\alpha}\in\mathbb{N}$ where $\alpha\in\Sigma^{+}_{\overline{0}}$,
$\lambda|_{\mathfrak{h}\cap\mathfrak{k}}=0$, and $\lambda$ is high enough,
then $V(\lambda,\mathfrak{b})$ is spherical.

Let $\mathfrak{n}^{-}$ be the nilpotent subalgebra for
$\Sigma^{-}:=-\Sigma^{+}$ with the corresponding Weyl vector $\rho^{-}=-\rho$.
We may then write down the opposite Iwasawa decomposition

(3) $\mathfrak{g}=\mathfrak{n}^{-}\oplus\mathfrak{a}\oplus\mathfrak{k}.$

From now on, we set

(4)
${\Gamma}:={\Gamma}_{\rho^{-}},\quad\gamma^{\mathtt{0}}:=\gamma^{\mathtt{0}}_{\rho^{-}}.$

The following Lemma is [Zhu22, Lemma 5.1].

######  Lemma 2.1.

Let $V$ be a $\mathfrak{g}$-module. If $\dim V^{\mathfrak{k}}=1$, then
$D\in\mathfrak{U}^{\mathfrak{k}}$ acts on $V^{\mathfrak{k}}$ by a scalar.

A result of the second author [Zhu22, Proposition 4.1] says that any finite
dimensional irreducible $\mathfrak{k}$-spherical $\mathfrak{g}$-module has a
spherical vector unique up to constant. Another result in the same work
[Zhu22, Theorem 5.2] relates this scalar with the highest weight restricted on
$\mathfrak{a}$.

######  Theorem 2.2.

If $V=V(\lambda,\mathfrak{b})$ is spherical then $\dim V^{\mathfrak{k}}=1$,
and $D\in\mathfrak{U}^{\mathfrak{k}}$ acts on $V^{\mathfrak{k}}$ by the scalar
given by ${\Gamma}(D)\left(\lambda|_{\mathfrak{a}}+\rho\right)$.

###  2.2. The Harish-Chandra homomorphism for
$\mathfrak{Z}(\mathfrak{gl}(m|n))$

Let $\mathfrak{g}=\mathfrak{gl}(m|n)$, and $\mathfrak{t}$ be the standard
Cartan subalgebra. We denote the root system of $\mathfrak{g}$ with respect to
$\mathfrak{t}$ as $\Sigma(\mathfrak{g},\mathfrak{t})$, and fix a choice of
positive system $\Sigma^{+}(\mathfrak{g},\mathfrak{t})$ in
$\Sigma(\mathfrak{g},\mathfrak{t})$. Let $\mathfrak{N}^{+}$ and
$\mathfrak{N}^{-}$ be the sums of positive root spaces and negative root
spaces respectively. Then
$\mathfrak{g}=\mathfrak{N}^{-}\oplus\mathfrak{t}\oplus\mathfrak{N}^{+}$ is the
triangular decomposition of $\mathfrak{g}$. We denote the Weyl vector
$\frac{1}{2}\sum_{\alpha\in\Sigma^{+}(\mathfrak{g},\mathfrak{t})}m_{\alpha}\alpha$
for $\Sigma^{+}(\mathfrak{g},\mathfrak{t})$, as $\rho_{\mathfrak{t}}$, where
$m_{\alpha}$ is the multiplicity of
$\alpha\in\Sigma^{+}(\mathfrak{g},\mathfrak{t})$.

We denote the universal enveloping algebra $\mathfrak{U}(\mathfrak{g})$ as
$\mathfrak{U}$, and the center of $\mathfrak{U}$ as $\mathfrak{Z}$. Following
[Hum78, Mus12], by the Poincaré–Birkhoff–Witt theorem, we have

(5)
$\mathfrak{U}=\mathfrak{S}(\mathfrak{t})\oplus(\mathfrak{N}^{-}\mathfrak{U}+\mathfrak{U}\mathfrak{N}^{+})$

Let $\xi$ be the respective projection onto $\mathfrak{S}(\mathfrak{t})$. On
$\mathfrak{S}(\mathfrak{t})$, we define an automorphism $\eta$ so that
$(\lambda+\rho_{\mathfrak{t}})(\eta(H))=\lambda(H)$ for all
$H\in\mathfrak{S}(\mathfrak{t})$. This automorphism can be equivalently
extended from $\eta(h):=h-\rho_{\mathfrak{t}}(h)$ for $h\in\mathfrak{t}$. The
Harish-Chandra isomorphism is then defined as

$\gamma_{\mathfrak{t}}:=\eta\circ\xi|_{\mathfrak{Z}}.$

Denote by $\chi_{\lambda}$ the central character afforded by a
$\mathfrak{g}$-module of highest weight $\lambda\in\mathfrak{t}^{*}$. Then

(6)
$\chi_{\lambda}(z)=\xi(z)=(\lambda+\rho_{\mathfrak{t}})\gamma_{\mathfrak{t}}(z)=\gamma_{\mathfrak{t}}(z)(\lambda+\rho_{\mathfrak{t}}).$

The last equality is due to identification between
$\mathfrak{P}(\mathfrak{t}^{*})$ and $\mathfrak{S}(\mathfrak{t})$.

To describe the image of $\gamma_{\mathfrak{t}}$, we first introduce the
supersymmetric polynomials. Let $\\{x_{i}\\}:=\\{x_{i}\\}_{i=1}^{m}$ and
$\\{y_{j}\\}:=\\{y_{j}\\}_{j=1}^{n}$ be two sets of independent variables, and
set $\\{x_{i},y_{j}\\}:=\\{x_{i}\\}\cup\\{y_{j}\\}$. We write
$f(x_{1}=t,y_{1}=-t)$ for the polynomial obtained by the substitutions
$x_{1}=t$ and $y_{1}=-t$. A polynomial $f$ in $\\{x_{i},y_{j}\\}$ is said to
be supersymmetric if

  1. (1)

$f$ is invariant under permutations of $\\{x_{i}\\}$ and of $\\{y_{j}\\}$
separately.

  2. (2)

$f(x_{1}=t,y_{1}=-t)$ is independent of $t$.

Let $I_{\mathbb{C}}(x_{i},y_{j})$ denote the $\mathbb{C}$-algebra of
supersymmetric polynomials in $\\{x_{i},y_{j}\\}$. By [Ste85, Theorem 1] (c.f.
[Mus12, Theorem 12.4.1]), $I_{\mathbb{C}}(x_{i},y_{j})$ is generated by the
power sums:

(7)
$p_{r}^{(m,n)}(x_{i},y_{j}):=\sum_{i=1}^{m}x_{i}^{r}-(-1)^{r}\sum_{j=1}^{n}y_{j}^{r},\;r\in\mathbb{Z}_{>0}.$

We record the following standard result about $\gamma$ where we identify
$\\{x_{i},y_{j}\\}$ as standard basis for $\mathfrak{t}$. Thus $x_{i}=E_{i,i}$
for $1\leq i\leq m$, and $y_{j}=E_{m+j,m+j}$ for $1\leq j\leq n$. We refer to
[Mus12, Theorem 13.1.1, Theorem 13.4.1], which are based on the original works
by Kac, Serganova, and Gorelik [Kac84, Ser99, Gor04].

######  Theorem 2.3.

The homomorphism $\gamma_{\mathfrak{t}}$ is an isomorphism and
$\operatorname{Im}\gamma_{\mathfrak{t}}=I_{\mathbb{C}}(x_{i},y_{j})$.

We also use the notation $\Lambda(\mathfrak{t}^{*})$ for the algebra of
supersymmetric polynomials on $\mathfrak{t}^{*}$ when we suppress the choice
of coordinates. Then we have:

(8)
$\operatorname{Im}\gamma_{\mathfrak{t}}=\Lambda(\mathfrak{t}^{*}),\quad\mathfrak{Z}\xrightarrow[\sim]{\gamma}\Lambda(\mathfrak{t}^{*}).$

######  Remark 1.

The above result is true in a more general term. In particular, Eq. (8) is
true when [Mus12, Hypothesis 8.3.4] is assumed, that is, when $\mathfrak{g}$
is $\mathfrak{gl}$, or basic simple, excluding type $A(m,n)$.

###  2.3. Cheng–Wang decomposition

Let $\mathfrak{k}=\mathfrak{gl}(m|n)\oplus\mathfrak{gl}(m|n)$. We recall a
multiplicity-free $\mathfrak{k}$-module decomposition, known as Howe duality
in [CW01], which generalizes Schmid’s decomposition [Sch70, FK90]. It will
allow us to define the supersymmetric Shimura operators in Section 3.

We first introduce some notation. A partition $\lambda$ is a sequence of non-
negative integers $(\lambda_{1},\lambda_{2},\dots)$ with only finitely many
non-zero terms and $\lambda_{i}\geq\lambda_{i+1}$ (c.f. [Mac95]). Let
$|\lambda|:=\sum_{i}\lambda_{i}$ denote the size of $\lambda$,
$\ell(\lambda):=\max\\{i:\lambda_{i}>0\\}$ the length of $\lambda$, and
$\lambda^{\prime}$ for which $\lambda^{\prime}_{i}:=|\\{j:\lambda_{j}\geq
i\\}|$ the transpose of $\lambda$. When viewed as the corresponding Young
diagram, $\lambda$ is the collection of “boxes” $(i,j)$

$\\{(i,j):1\leq i\leq\ell(\lambda),1\leq j\leq\lambda_{i}\\}.$

A $(m,n)$-hook partition is a partition $\lambda$ such that $\lambda_{m+1}\leq
n$. We define

$\mathscr{H}(m,n):=\left\\{\lambda:\lambda_{m+1}\leq
n\right\\},\quad\mathscr{H}^{d}(m,n):=\left\\{\lambda\in\mathscr{H}(m,n):|\lambda|=d\right\\}.$

For $\lambda\in\mathscr{H}(m,n)$, we define a $(m+n)$-tuple

(9)
$\lambda^{\natural}:=\left(\lambda_{1},\dots,\lambda_{m},\left\langle\lambda_{1}^{\prime}-m\right\rangle,\dots,\left\langle\lambda_{n}^{\prime}-m\right\rangle\right)$

where $\langle x\rangle:=\max\\{x,0\\}$ for $x\in\mathbb{Z}$. The last $n$
coordinates can be viewed as the lengths of the remaining columns after
discarding the first $m$ rows of $\lambda$.

Let $\mathfrak{B}$ be any Borel subalgebra of $\mathfrak{gl}(m|n)$ containing
a Cartan subalgebra $\mathfrak{H}\subseteq\mathfrak{gl}(m|n)_{\overline{0}}$.
Then $\mathfrak{B}$ can be described by an $\epsilon\delta$-chain,
$[X_{1}\cdots X_{m+n}]$, a sequence consisting of characters
$X_{i}\in\mathfrak{H}^{*}$ where $X_{i}-X_{i+1}$ exhaust all the simple roots
defining $\mathfrak{B}$. Therefore,

$[\epsilon_{1}\cdots\epsilon_{m}\delta_{1}\cdots\delta_{n}]$

gives the standard Borel subalgebra $\mathfrak{b}^{\operatorname{st}}$, while
$[\delta_{n}\cdots\delta_{1}\epsilon_{m}\cdots\epsilon_{1}]$ gives the
opposite one, denoted as $\mathfrak{b}^{\operatorname{op}}$. For an
$(m+n)$-tuple $(a_{1},\dots,a_{m},b_{1},\dots,b_{n})$, we associate an
irreducible $\mathfrak{gl}(m|n)$-module of highest weight
$\sum_{i=1}^{m}a_{i}\epsilon_{i}+\sum_{j=1}^{n}b_{j}\delta_{j}$ with respect
to $\mathfrak{b}^{\operatorname{st}}$, denoted as
$L(a_{1},\dots,a_{m},b_{1},\dots,b_{n})$.

Let $V$ be a vector superspace and $\mathfrak{S}(V)$ be the supersymmetric
algebra on $V$, so $\mathfrak{S}(V)$ has a natural $\mathbb{N}$-grading
$\bigoplus_{k\in\mathbb{N}}\mathfrak{S}^{n}(V)$, and explicitly,
$\mathfrak{S}^{d}(V)=\bigoplus_{i+j=d}\mathfrak{S}^{i}(V_{\overline{0}})\otimes\bigwedge^{j}(V_{\overline{1}})$
as vector spaces. The natural action of $\mathfrak{gl}(m|n)$ on
$\mathbb{C}^{m|n}$ gives an action of
$\mathfrak{k}=\mathfrak{gl}(m|n)\oplus\mathfrak{gl}(m|n)$ on
$\mathbb{C}^{m|n}\otimes\mathbb{C}^{m|n}$, which extends to an action on
$\mathfrak{S}(\mathbb{C}^{m|n}\otimes\mathbb{C}^{m|n})$. We record the
following result regarding the $\mathfrak{k}$-module structure on
$\mathfrak{S}(\mathbb{C}^{m|n}\otimes\mathbb{C}^{m|n})$. See [CW01, Theorem
3.2].

######  Theorem 2.4.

The supersymmetric algebra
$\mathfrak{S}(\mathbb{C}^{m|n}\otimes\mathbb{C}^{m|n})$ as a
$\mathfrak{k}$-module is completely reducible and multiplicity-free. In
particular,

$\mathfrak{S}^{d}(\mathbb{C}^{m|n}\otimes\mathbb{C}^{m|n})=\bigoplus_{\lambda\in\mathscr{H}^{d}(m,n)}L(\lambda^{\natural})\otimes
L(\lambda^{\natural}).$

###  2.4. Supersymmetric Polynomials

We devote this subsection to an overview of known results of supersymmetric
polynomials, including the Type BC interpolation polynomials introduced and
studied by Sergeev and Veselov in [SV09]. These are super analog for the now-
classic Okounkov polynomials [Oko98, OO06]. We specialize them to $J_{\mu}$
(as in Theorem A) in Section 6.

Let $\\{\alpha^{\textsc{F}}_{j}\\}\cup\\{\alpha^{\textsc{B}}_{i}\\}$ be the
standard basis for $\mathbb{C}^{m+n}$, and $w_{j}$ and $z_{i}$ be the
coordinate functions of $\alpha^{\textsc{F}}_{j}$ and
$\alpha^{\textsc{B}}_{i}$, for $j=1,\dots,m,i=1\dots,n$. Let $\mathsf{k}$ and
$\mathsf{h}$ be two parameters. Following [SV09], we assume that
$\mathsf{k}\notin\mathbb{Q}_{>0}$ (generic), and set $\hat{\mathsf{h}}$ by
$2\hat{\mathsf{h}}-1=\mathsf{k}^{-1}(2\mathsf{h}-1)$. We also set

$\displaystyle C_{\lambda}^{+}(x;\mathsf{k})$
$\displaystyle:=\prod_{(i,j)\in\lambda}\left(\lambda_{i}+j+\mathsf{k}(\lambda^{\prime}_{j}+i)+x\right)$
$\displaystyle C_{\lambda}^{-}(x;\mathsf{k})$
$\displaystyle:=\prod_{(i,j)\in\lambda}\left(\lambda_{i}-j-\mathsf{k}(\lambda^{\prime}_{j}-i)+x\right)$

As in [SV09, Section 6], we let
$\rho:=\sum\rho_{j}^{\textsc{F}}\alpha^{\textsc{F}}_{j}+\sum\rho_{i}^{\textsc{B}}\alpha^{\textsc{B}}_{i}$
where

(10)
$\rho_{j}^{\textsc{F}}:=-(\mathsf{h}+\mathsf{k}j+n),\;\rho_{i}^{\textsc{B}}:=-\mathsf{k}^{-1}\left(\mathsf{h}+\frac{1}{2}\mathsf{k}-\frac{1}{2}+i\right).$

This is in fact the deformed Weyl vector calculated with the deformed root
multiplicities in [SV09]. We identify $P_{m,n}:=\mathbb{C}[w_{j},z_{i}]$ with
$\mathfrak{P}(\mathbb{C}^{m+n})$, and define
$\Lambda^{(\mathsf{k},\mathsf{h})}_{m,n}\subseteq P_{m,n}$ as the subalgebra
of polynomials $f$ which: (1) are symmetric separately in shifted variables
$(w_{j}-\rho_{j}^{\textsc{F}})$ and $(z_{i}-\rho_{i}^{\textsc{B}})$, and
invariant under their sign changes; (2) satisfy the condition

$f(X-\alpha^{\textsc{B}}_{i}+\alpha^{\textsc{F}}_{j})=f(X)$

on the hyperplane $w_{j}+\mathsf{k}(j-1)=\mathsf{k}z_{i}+i-n$. If we equip
$\mathbb{C}^{m+n}$ with an inner product defined by

$(\alpha^{\textsc{F}}_{m},\alpha^{\textsc{F}}_{n})=\delta_{m,n},\;(\alpha^{\textsc{B}}_{m},\alpha^{\textsc{B}}_{n})=\mathsf{k}\delta_{m,n},\;(\alpha^{\textsc{F}}_{j},\alpha^{\textsc{B}}_{i})=0$

Then Condition (2) is equivalent to: (2’) $f(X+\alpha)=f(X)$ for any
$\alpha=\alpha^{\textsc{B}}_{i}\pm\alpha^{\textsc{F}}_{j}$ on the hyperplane

$(X-\rho,\alpha)+\frac{1}{2}(\alpha,\alpha)=0.$

For $\lambda\in\mathscr{H}(m,n)$, we set
$w(\lambda)=(\lambda^{\prime}_{1},\dots,\lambda^{\prime}_{n})$ and
$z(\lambda)=(\left\langle\lambda_{1}-n\right\rangle,\dots,\left\langle\lambda_{m}-n\right\rangle)$.
Equivalently, $(z(\lambda),w(\lambda))=(\lambda^{\prime})^{\natural}$ (see Eq.
(9)). Then [SV09, Proposition 6.3] says the following.

######  Theorem 2.5.

For each $\mu\in\mathscr{H}(m,n)$, there exists a unique polynomial
$I^{\textup{SV}}_{\mu}(X;\mathsf{k,h})\in\Lambda^{(\mathsf{k},\mathsf{h})}_{n,m}$
of degree $2|\mu|$ such that

$I^{\textup{SV}}_{\mu}(z(\lambda),w(\lambda);\mathsf{k,h})=0,\quad\text{for
all }\lambda\nsupseteq\mu,\,\lambda\in\mathscr{H}(m,n).$

and satisfies the normalization condition
$I^{\textup{SV}}_{\mu}(w(\mu),z(\mu);\mathsf{k,h})=C_{\mu}^{-}(1;\mathsf{k})C_{\mu}^{+}(2\mathsf{h}-1;\mathsf{k})$.
Moreover, $\\{I^{\textup{SV}}_{\mu}\\}$ is a basis for $\Lambda^{\rho}$.

In the same work [SV09], a closely related polynomial is introduced

(11)
$I^{\textup{SV}}_{\mu}(z_{i},w_{j};\mathsf{k},\mathsf{h}):=d_{\mu}\hat{I}_{\mu^{\prime}}(z_{i},w_{j};\mathsf{k}^{-1},\hat{\mathsf{h}}),$

for

(12)
$d_{\mu}:=(-1)^{|\mu|}\mathsf{k}^{2|\mu|}\frac{C_{\mu}^{-}(1;\mathsf{k})}{C_{\mu}^{-}(-\mathsf{k};\mathsf{k})}.$

A tableau formula is provided in [SV09, Proposition 6.4] as follows.

######  Theorem 2.6.

For $\lambda\in\mathscr{H}(m,n)$, we have

(13)
$\hat{I}_{\lambda}(z_{i},w_{j};\mathsf{k},\mathsf{h})=\sum_{T}\varphi_{T}(-\mathsf{k})\prod_{(i,j)\in\lambda}f_{T}(i,j).$

Here $T$ is any reverse bitableau of type $(n,m)$ and shape
$\lambda\in\mathscr{H}(m,n)$, with a filling by symbols
$1<\cdots<n<1^{\prime}<\cdots<m^{\prime}$ (see [SV09, Section 6]). The weight
$\varphi_{T}$ is defined as in [SV05, Eq. (41)], and $f_{T}(i,j)$ has leading
term $z_{a}^{2}$ if $T(i,j)=a$ for $a=1,\dots,n$ and leading term
$\mathsf{k}^{2}w_{b}^{2}$ if $T(i,j)=b^{\prime}$ for
$b^{\prime}=1^{\prime},\dots,m^{\prime}$. In fact, [SV09] uses Eq. (13) as the
definition for $\hat{I}_{\lambda}$ and Eq. (11) as a proposition. Here we
present the tableau formula as a theorem in parallel with the following result
regarding super Jack polynomials.

We recall the theory of super Jack polynomials from [SV05]. The (monic) Jack
symmetric functions $P_{\lambda}(x;\theta)$ are a linear basis for the ring
$\Lambda$ of symmetric functions, and the power sums $p_{r}$ are free
generators $\Lambda$, see e.g. [Mac95]. Let $\phi_{\theta}$ be the
homomorphism from $\Lambda$ to the polynomial ring $\mathbb{C}[x_{i},y_{j}]$
which is defined on the generators as follows

$\phi_{\theta}(p_{r})=p^{(m,n)}_{r,\theta}(x_{i},y_{j}):=\sum_{i=1}^{m}x_{i}^{r}-\frac{1}{\theta}\sum_{j=1}^{n}y_{j}^{r}.$

Then the super Jack polynomial $SP_{\lambda}$ is defined by

$SP_{\lambda}=SP_{\lambda}(x_{i},y_{j};\theta):=\phi_{\theta}(P_{\lambda}(x;\theta)).$

######  Theorem 2.7.

We have

(14)
$SP_{\lambda}(x_{i},y_{j};\theta)=\sum_{T}\varphi_{T}(\theta)\prod_{(i,j)\in\lambda}x_{T(i,j)}$

where $x_{j^{\prime}}$ is $y_{j}$ for $j=1,\dots,n$ and for $\varphi_{T}$
defined as in [SV05, Eq. (41)].

We also record the following expansion ([Yan92, Sta89]) as in [SZ19] (note
$\tau$ is $\theta$ here)

(15)
$\frac{1}{m!}\left(\sum_{i}x_{i}^{2}\right)^{m}=\sum_{|\mu|=m}C_{\mu}^{-}(1;-\theta)P_{\mu}(x_{i}^{2};\theta).$

Applying $\phi_{\theta}$ on both sides, we have

(16)
$\frac{1}{m!}\left(p_{2,\theta}^{(m,n)}(x_{i},y_{j};\theta)\right)^{m}=\sum_{\mu\in\mathscr{H}^{m}(m,n)}C_{\mu}^{-}(1;-\theta)SP_{\mu}(x_{i}^{2},y_{j}^{2};\theta).$

##  3\. Supersymmetric Shimura Operators

From now on, we fix $\mathfrak{g}=\mathfrak{gl}(2p|2q)$ and
$\mathfrak{k}=\mathfrak{gl}(p|q)\oplus\mathfrak{gl}(p|q)$. This section is
devoted to the description of the pair $(\mathfrak{g},\mathfrak{k})$ and
important subspaces therein. We then define the supersymmetric Shimura
operators and specialize related results introduced above. Throughout the
section we set indices $i=1,\dots,p,j=1\dots,q$, and $k$ is either $i$ or $j$
depending on the context.

###  3.1. Realization

We fix the following embedding of $\mathfrak{k}$ into $\mathfrak{g}$, and
identify $\mathfrak{k}$ with its image

(17) $\left(\left(\begin{array}[]{c|c}A_{p\times p}&B_{p\times q}\\\ \hline\cr
C_{q\times p}&D_{q\times
q}\end{array}\right),\left(\begin{array}[]{c|c}A_{p\times
p}^{\prime}&B_{p\times q}^{\prime}\\\ \hline\cr C_{q\times
p}^{\prime}&D_{q\times
q}^{\prime}\end{array}\right)\right)\mapsto\left(\begin{array}[]{c c |c c
}A_{p\times p}&0_{p\times p}&B_{p\times q}&0_{p\times q}\\\ 0_{p\times
p}&A_{p\times p}^{\prime}&0_{p\times q}&B_{p\times q}^{\prime}\\\ \hline\cr
C_{q\times p}&0_{q\times p}&D_{q\times q}&0_{q\times q}\\\ 0_{q\times
p}&C_{q\times p}^{\prime}&0_{q\times q}&D_{q\times
q}^{\prime}\end{array}\right)$

We let $J:=\frac{1}{2}\operatorname{diag}(I_{p\times p},-I_{p\times
p},I_{q\times q},-I_{q\times q})$, and $\theta:=\operatorname{Ad}\exp(i\pi
J)$. Then $\theta$ has fixed point subalgebra
$\mathfrak{k}\cong\mathfrak{gl}(p|q)\oplus\mathfrak{gl}(p|q)$. We also have
the Harish-Chandra decomposition

$\mathfrak{g}=\mathfrak{p}^{-}\oplus\mathfrak{k}\oplus\mathfrak{p}^{+}$

where $\mathfrak{p}^{+}$ (respectively $\mathfrak{p}^{-}$) consists of
matrices with non-zero entries only in the upper right (respectively bottom
left) sub-blocks in each of the four blocks. Set
$\mathfrak{p}:=\mathfrak{p}^{-}\oplus\mathfrak{p}^{+}$. Then
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$.

In our theory, we need to work with a $\theta$-stable Cartan subalgebra
$\mathfrak{h}$ of $\mathfrak{g}$ extended from the toral subalgebra
$\mathfrak{a}\subseteq\mathfrak{p}_{\overline{0}}$. Note in [AHZ10, Section
4], it is showed that
$(\mathfrak{gl}(r+p|s+q),\mathfrak{gl}(r|s)\oplus\mathfrak{gl}(p|q))$ is of
even type if and only if $(r-p)(s-q)\geq 0$, satisfied by our pair with
$p=r,q=s$. We present a construction of $\mathfrak{h}$ and $\mathfrak{a}$
using a certain Cayley transform as follows.

We let $\epsilon_{i}^{+}:=\epsilon_{i}$, $\epsilon_{i}^{-}:=\epsilon_{p+i}$,
$\delta_{j}^{+}:=\delta_{j}$, and $\delta_{j}^{-}:=\delta_{q+j}$ be the
characters on $\mathfrak{t}$. Let

$\gamma_{i}^{\textsc{B}}:=\epsilon_{i}^{+}-\epsilon_{i}^{-},\quad\gamma_{j}^{\textsc{F}}:=\delta_{j}^{+}-\delta_{j}^{-}.$

These are the Harish-Chandra strongly orthogonal roots, and we denote the set
of $\gamma_{k}^{\textsc{B/F}}$ 111Here B indicates the Boson–Boson block (top
left) and F the Fermion–Fermion block (bottom right), c.f. [All12]. as
$\Sigma_{\perp}$. We set $D_{j,j^{\prime}}:=E_{2p+j,2p+j^{\prime}}$ for $1\leq
j,j^{\prime}\leq 2q$. Associated with each $\gamma^{\textsc{B}}_{i}$ is an
$\mathfrak{sl}(2)$-triple spanned by $E_{i,i}-E_{p+i,p+i},E_{i,p+i}$ and
$E_{p+i,i}$ (similarly for $\gamma^{\textsc{F}}_{j}$ with $D_{j,j^{\prime}}$).
It is not hard to see that all $(p+q)$ $\mathfrak{sl}(2)$-triples commute. We
write $\mathsf{i}$ for the imaginary unit $\sqrt{-1}$ to avoid confusion.
Define

$c^{\textsc{B}}_{i}:=\operatorname{Ad}\exp\left({\frac{\pi}{4}}\mathsf{i}(-E_{i,p+i}-E_{p+i,i})\right),\quad
c^{\textsc{F}}_{j}:=\operatorname{Ad}\exp\left({\frac{\pi}{4}}\mathsf{i}(-D_{j,q+j}-D_{q+j,j})\right)$

The product

(18) $c:=\prod_{i}c^{\textsc{B}}_{i}\prod_{j}c^{\textsc{F}}_{j}$

is thus a well-defined automorphism on $\mathfrak{g}$ as all terms commute. We
set

$\displaystyle x_{i}$ $\displaystyle:=\mathsf{i}(E_{i,p+i}-E_{p+i,i}),$
$\displaystyle\quad x^{\prime}_{i}$ $\displaystyle:=E_{i,i}+E_{p+i,p+i},$
$\displaystyle\quad x_{\pm i}$ $\displaystyle:=\frac{1}{2}(x^{\prime}_{i}\pm
x_{i})$ $\displaystyle y_{j}$
$\displaystyle:=\mathsf{i}(D_{j,q+j}-D_{q+j,j}),$ $\displaystyle\quad
y^{\prime}_{j}$ $\displaystyle:=D_{j,j}+D_{q+j,q+j},$ $\displaystyle\quad
y_{\pm j}$ $\displaystyle:=\frac{1}{2}(y^{\prime}_{j}\pm y_{j})$

Then by a direct (rank 1) computation, we see that under $c$:

(19) $E_{i,i}\mapsto x_{+i},\quad E_{p+i,p+i}\mapsto x_{-i},\quad
D_{j,j}\mapsto y_{+j},\quad D_{q+j,q+j}\mapsto y_{-j}.$

We now define

(20)
$\mathfrak{h}:=c(\mathfrak{t}),\quad\mathfrak{a}:=\operatorname{Span}_{\mathbb{C}}\\{x_{i},y_{j}\\},\quad\mathfrak{t}_{+}:=\operatorname{Span}_{\mathbb{C}}\\{x^{\prime}_{i},y^{\prime}_{j}\\}.\quad(\text{Note
}\mathfrak{h}=\mathfrak{a}\oplus\mathfrak{t}_{+}.)$

In $\mathfrak{t}$, the space
$\mathfrak{t}_{-}:=\operatorname{Span}_{\mathbb{C}}\\{E_{i,i}-E_{p+i,p+i},D_{j,j}-D_{q+j,q+j}\\}$
is the orthogonal complement of $\mathfrak{t}_{+}$ with respect to the Killing
form on $\mathfrak{g}$. Also, on $\mathfrak{h}$ we let

$\alpha^{\textsc{B}}_{i},\alpha^{\textsc{F}}_{j},\tau_{i}^{\textsc{B}},\tau_{j}^{\textsc{F}}\in\mathfrak{h}^{*}$

be dual to $x_{i},y_{j},x^{\prime}_{i},y^{\prime}_{j}\in\mathfrak{h}$
respectively. Then $\tau_{k}^{\textsc{B/F}}$ and $\alpha_{k}^{\textsc{B/F}}$
vanish on $\mathfrak{a}$ and $\mathfrak{t}_{+}$ respectively. We identify
$\alpha_{k}^{\textsc{B/F}}$ with its restriction to $\mathfrak{a}$. We also
have $\tau_{i}^{\textsc{B}}=\frac{1}{2}(\epsilon_{i}^{+}+\epsilon_{i}^{-})$,
$\tau_{j}^{\textsc{F}}=\frac{1}{2}(\delta_{j}^{+}+\delta_{j}^{-})$ and we
identify them with their restrictions on $\mathfrak{t}_{+}$. On
$\mathfrak{a}^{*}$, we set

(21)
$(\alpha_{m}^{\textsc{B}},\alpha_{n}^{\textsc{B}})=-(\alpha_{m}^{\textsc{F}},\alpha_{n}^{\textsc{F}})=\delta_{mn},\quad(\alpha_{i}^{\textsc{B}},\alpha_{j}^{\textsc{F}})=0,$

which is induced from $b$, the one-half of the supertrace form. For future
purposes, we also consider the following basis for $\mathfrak{h}^{*}$:

(22) $\chi_{\pm
i}:=\tau_{i}^{\textsc{B}}\pm\alpha_{i}^{\textsc{B}},\quad\eta_{\pm
j}:=\tau_{j}^{\textsc{F}}\pm\alpha_{j}^{\textsc{F}}.$

###  3.2. Root Data

Recall the superdimension of a super vector space $V$ is denoted as
$\operatorname{sdim}V=\dim V_{\overline{0}}|\dim V_{\overline{1}}$. Recall
that for a (restricted) root space $\mathfrak{g}_{\alpha}$ of
$\alpha\in\Sigma(\mathfrak{g},\cdot)$, the root multiplicity is defined as
$m_{\alpha}:=\dim(\mathfrak{g}_{\alpha})_{\overline{0}}-\dim(\mathfrak{g}_{\alpha})_{\overline{1}},$
and the deformed root multiplicity is defined as

$\operatorname{\mathit{m}}(\alpha):=-\frac{1}{2}m_{\alpha}.$

We first give an explicit description of $\Sigma(\mathfrak{g},\mathfrak{h})$
using the coordinates defined in Eq. (22). The order $<$ on the indices
$\\{\pm i,\pm j\\}$ is interpreted as the natural order on $\mathbb{Z}$, where
$+i$ is identified with $i$. Then we have

(23) $\displaystyle\Sigma_{\overline{0}}^{+}(\mathfrak{g},\mathfrak{h})$
$\displaystyle=\\{\chi_{k}-\chi_{l}:k<l\\}\cup\\{\eta_{k}-\eta_{l}:k<l\\},\quad$
$\displaystyle m_{\alpha}=1$ (24)
$\displaystyle\Sigma_{\overline{1}}^{+}(\mathfrak{g},\mathfrak{h})$
$\displaystyle=\\{\chi_{k}-\eta_{l}:+1\leq k\leq+p,-q\leq l\leq+q\\}$
$\displaystyle\cup\\{\eta_{l}-\chi_{k}:-q\leq l\leq+q,-p\leq k\leq-1\\},\quad$
$\displaystyle m_{\alpha}=-1$

This can be seen by (I) viewing $\mathfrak{h}$ as a Weyl group conjugate of
the standard Cartan subalgebra $\mathfrak{t}$ and identifying the roots
expressed using $\epsilon^{\pm}$ and $\delta^{\pm}$, or (II) considering the
Cayley transform which in turn gives a correspondence between positive roots
on $\mathfrak{h}$ and those on $\mathfrak{t}$. Alternatively, the chain in Eq.
(35) gives the above choice of positive roots.

We now record restricted root data taken from [Zhu22]. The restricted root
system $\Sigma(\mathfrak{g},\mathfrak{a})$ is of Type C. In the following
table, we give the standard choice of positivity, together with the
superdimensions of the restricted root spaces. The parameters
$\mathsf{k,p,q,r,s}$ are $\operatorname{\mathit{m}}(\alpha)$, and these are
the 5 parameters first introduced by Sergeev and Veselov in [SV09] to study
interpolation polynomials. Note only the last column gives the odd restricted
roots.

$\alpha^{\textsc{B}}_{i},\mathsf{r}$ |  $2\alpha^{\textsc{B}}_{i},\mathsf{s}$ |  $\alpha^{\textsc{B}}_{i}\pm\alpha^{\textsc{B}}_{k},\mathsf{k}$ |  $\alpha^{\textsc{F}}_{j},\mathsf{p}$ |  $2\alpha^{\textsc{F}}_{j},\mathsf{q}$ |  $\alpha^{\textsc{F}}_{j}\pm\alpha^{\textsc{F}}_{l},\mathsf{k}^{-1}$ |  $\alpha^{\textsc{B}}_{i}\pm\alpha^{\textsc{F}}_{j},1$
---|---|---|---|---|---|---
/ |  $1|0,-\frac{1}{2}$ |  $2|0,-1$ |  / |  $1|0,-\frac{1}{2}$ |  $2|0,-1$ |  $0|2,1$
Table 1. Positive Restricted Roots and $\operatorname{\mathit{m}}(\alpha)$.
$1\leq i<k\leq p,1\leq j<l\leq q$

Note their $\epsilon_{i}$ and $\delta_{p}$ are our $\alpha^{\textsc{F}}_{j}$
and $\alpha^{\textsc{B}}_{i}$ respectively, due to a different choice of
positivity (see [SV09, Eq. (71)]). A direct computation using Eqs. (23, 24)
shows the Weyl vector in $\mathfrak{h}^{*}$ is

$\displaystyle\rho_{\mathfrak{h}}$
$\displaystyle=\sum_{i=1}^{p}\left((p-i)+\frac{1}{2}-q\right)(\chi_{+i}-\chi_{-i})+\sum_{j=1}^{q}\left((q-i)+\frac{1}{2}\right)(\eta_{+j}-\eta_{-j})$
(25)
$\displaystyle=\sum_{i=1}^{p}(2(p-i)+1-2q)\alpha^{\textsc{B}}_{i}+\sum_{j=1}^{q}(2(q-j)+1)\alpha^{\textsc{F}}_{j}.$

The Weyl vector of restricted roots in $\mathfrak{a}^{*}$ is:

(26)
$\rho=\sum_{i=1}^{p}(2(p-i)+1-2q)\alpha^{\textsc{B}}_{i}+\sum_{j=1}^{q}(2(q-j)+1)\alpha^{\textsc{F}}_{j},$

as the restriction of $\rho_{\mathfrak{h}}$ on $\mathfrak{a}$.

###  3.3. Supersymmetric Shimura operators

Throughout the subsection, we set $i=1,\dots,p$ and $j=1,\dots,q$. Recall
$L(\lambda^{\natural})$ is the irreducible $\mathfrak{gl}(p|q)$-module of
highest weight

$\sum\lambda_{i}\epsilon_{i}+\sum\langle\lambda^{\prime}_{j}-p\rangle\delta_{j}$

with respect to $\mathfrak{b}^{\operatorname{st}}$, and it is of Type M
([CW12]). In this case, Schur’s Lemma indicates that
$\dim\operatorname{End}_{\mathfrak{k}}\left(L(\lambda^{\natural})\right)=1$,
and $L(\lambda^{\natural})\otimes L(\lambda^{\natural})$ is actually
irreducible as a $\mathfrak{k}$-module as
$\mathfrak{k}=\mathfrak{gl}(p|q)\oplus\mathfrak{gl}(p|q)$. If we let
$\mathfrak{gl}(p|q)$ act on the second component contragrediently (via a
negative supertranspose), then we define the irreducible $\mathfrak{k}$-module
$L(\lambda^{\natural})\otimes L^{*}(\lambda^{\natural})$ as $W_{\lambda}$.
Note both $\mathfrak{p}^{-}$ and $\mathfrak{p}^{+}$ are $\mathfrak{k}$-modules
by the short grading. We identify $\mathfrak{p}^{-}$ as
$(\mathfrak{p}^{+})^{*}$ via the form $b$ also used for Eq. (21).

######  Proposition 3.1.

The symmetric superalgebras $\mathfrak{S}(\mathfrak{p}^{+})$ and
$\mathfrak{S}(\mathfrak{p}^{-})$ are completely reducible and multiplicity
free as $\mathfrak{k}$-modules. Specifically,

(27)
$\mathfrak{S}^{d}(\mathfrak{p}^{+})=\bigoplus_{\lambda\in\mathscr{H}^{d}(p,q)}W_{\lambda},\quad\mathfrak{S}^{d}(\mathfrak{p}^{-})=\bigoplus_{\lambda\in\mathscr{H}^{d}(p,q)}W_{\lambda}^{*}.$

###### Proof.

By duality, it suffices to show the first equation. First we have
$\mathfrak{p}^{+}\cong\mathbb{C}^{m|n}\otimes(\mathbb{C}^{m|n})^{*}$, by
identifying $\mathbb{C}^{m|n}$ and $(\mathbb{C}^{m|n})^{*}$ as spaces of
column and row vectors respectively. The contragredient $\mathfrak{k}$-module
structure on $(\mathbb{C}^{m|n})^{*}$ is obtained by applying the negative
supertranspose on $\mathfrak{gl}(p|q)$. Then Theorem 2.4 implies

$\mathfrak{S}^{d}(\mathfrak{p}^{+})=\bigoplus_{\lambda\in\mathscr{H}^{d}(p,q)}L(\lambda^{\natural})\otimes
L^{*}(\lambda^{\natural}),$

proving the claim, c.f. [SSS20, Theorem 1.4] and notations therein. ∎

We write down the highest weight of $W_{\lambda}$ with respect to the Borel
subalgebra
$\mathfrak{b}^{\operatorname{st}}\oplus\mathfrak{b}^{\operatorname{op}}$ of
$\mathfrak{k}$:

(28) $\displaystyle\mathfrak{t}^{*}\ni\lambda^{\natural}_{\mathfrak{t}}$
$\displaystyle:=\sum\lambda_{i}\epsilon^{+}_{i}+\sum\langle\lambda_{j}^{\prime}-p\rangle\delta^{+}_{j}-\sum\lambda_{i}\epsilon^{-}_{i}-\sum\langle\lambda_{j}^{\prime}-p\rangle\delta^{-}_{j}$
(29)
$\displaystyle=\sum\lambda_{i}\gamma_{i}^{\textsc{B}}+\sum\langle\lambda_{j}^{\prime}-p\rangle\gamma_{j}^{\textsc{F}},\quad\gamma_{i}^{\textsc{B}},\gamma_{j}^{\textsc{F}}\in\Sigma_{\perp}.$

The part with negative terms is indeed dominant since we take the opposite
Borel subalgebra for the second copy of $\mathfrak{gl}(p|q)$ in
$\mathfrak{k}=\mathfrak{gl}(p|q)\oplus\mathfrak{gl}(p|q)$.

As $\mathfrak{p}^{\pm}$ are supercommutative, the respective universal
enveloping algebras are just $\mathfrak{S}(\mathfrak{p}^{\pm})$. The direct
summand $W_{\lambda}^{*}\otimes W_{\lambda}$ embedded in
$\mathfrak{S}(\mathfrak{p}^{-})\otimes\mathfrak{S}(\mathfrak{p}^{+})$ is then
multiplied into $\mathfrak{U}$. We write $1_{\lambda}$ for the element in
$\left(W_{\lambda}^{*}\otimes W_{\lambda}\right)^{\mathfrak{k}}$ corresponding
to
$\mathrm{Id}_{W_{\lambda}}\in\operatorname{End}_{\mathfrak{k}}(W_{\lambda})$.
Let
$\mathfrak{D}:=\mathfrak{U}^{\mathfrak{k}}/(\mathfrak{U}\mathfrak{k})^{\mathfrak{k}}$.

######  Definition 3.2.

For each $\lambda\in\mathscr{H}(p,q)$, we let $D_{\lambda}$ be the image
corresponding to $1_{\lambda}$ under the composition of the multiplication and
the embedding

$\displaystyle\left(W_{\lambda}^{*}\otimes
W_{\lambda}\right)^{\mathfrak{k}}\hookrightarrow\left(\mathfrak{S}(\mathfrak{p}^{-})\otimes\mathfrak{S}(\mathfrak{p}^{+})\right)^{\mathfrak{k}}\rightarrow$
$\displaystyle\mathfrak{U}^{\mathfrak{k}}$
$\displaystyle\rightarrow\mathfrak{D}$ (30) $\displaystyle
1_{\lambda}\xmapsto{\hphantom{\left(W_{\lambda}^{*}\otimes
W_{\lambda}\right)^{\mathfrak{k}}\hookrightarrow\left(\mathfrak{S}(\mathfrak{p}^{-})\otimes\mathfrak{S}(\mathfrak{p}^{+})\right)^{\mathfrak{k}}\rightarrow\mathfrak{U}^{\mathfrak{k}}}}$
$\displaystyle D_{\lambda}$ $\displaystyle\mapsto\mathscr{D}_{\lambda}$

The element $\mathscr{D}_{\lambda}$ is called the supersymmetric Shimura
operator associated with the partition $\lambda$.

######  Remark 2.

For an irreducible Hermitian symmetric space $G/K$, such $D_{\lambda}$ can be
similarly defined (c.f. [SZ19]) in $\mathfrak{D}(G)$, the algebra of
differential operators on $G$, identified with the universal enveloping
algebra of the Lie algebra of $G$. As $D_{\lambda}$ commutes with $K$, its
right action descends to $\mathscr{D}_{\lambda}\in\mathfrak{D}$, the algebra
of differential operators on $G/K$. These $\mathscr{D}_{\lambda}$ are the
original Shimura operators. As we will study the action of $D_{\lambda}$ on
$\mathfrak{g}$-modules, we call $D_{\lambda}\in\mathfrak{U}^{\mathfrak{k}}$ as
(supersymmetric) Shimura operators as well, by a slight abuse of name. Working
with the lift $D_{\lambda}\in\mathfrak{U}^{\mathfrak{k}}$ of
$\mathscr{D}_{\lambda}\in\mathfrak{U}^{\mathfrak{k}}$ gives tremendous
flexibility. By the definitions of ${\Gamma}$, $\gamma^{\mathtt{0}}$,
$D_{\lambda}$ and $\mathscr{D}_{\lambda}$, we see

(31) $\gamma^{\mathtt{0}}(\mathscr{D}_{\lambda})={\Gamma}(D_{\lambda}).$

###  3.4. Specialized Results

We now specialize some of the results we introduced in Section 2. First, we
set the choice of positive roots $\Sigma^{+}(\mathfrak{g},\mathfrak{t})$
according to the chain (c.f. Subsection 2.3)

(32)
$[\epsilon_{1}^{+}\cdots\epsilon_{p}^{+}\delta_{1}^{+}\cdots\delta_{q}^{+}\delta_{q}^{-}\cdots\delta_{1}^{-}\epsilon_{p}^{-}\cdots\epsilon_{1}^{-}]$

which equivalently gives a Borel subalgebra of $\mathfrak{g}$, denoted as
$\mathfrak{b}^{\natural}$. In fact, we have

(33)
$\mathfrak{b}^{\natural}=\mathfrak{b}^{\operatorname{st}}\oplus\mathfrak{b}^{\operatorname{op}}\oplus\mathfrak{p}^{+}.$

Note the Cayley transform $c$ as in Eq. (18) allows us to send the positivity
on one Cartan subalgebra to that of the other Cartan subalgebra. Specifically,
we have $c^{-1}:\mathfrak{h}\rightarrow\mathfrak{t}$, and the dual map
$c^{-1}_{*}:\mathfrak{t}^{*}\rightarrow\mathfrak{h}^{*}$. It is then a direct
computation to check that

(34) $c^{-1}_{*}:\epsilon^{\pm}_{i}\mapsto\chi_{\pm
i},\quad\delta^{\pm}_{j}\mapsto\eta_{\pm j}.$

Thus, the induced choice of positivity of $\Sigma(\mathfrak{g},\mathfrak{h})$
is given by

(35)
$[\chi_{+1}\cdots\chi_{+p}\eta_{+1}\cdots\eta_{+q}\eta_{-q}\cdots\eta_{-1}\chi_{-p}\cdots\chi_{-1}].$

We denote the corresponding Borel subalgebra as $\mathfrak{b}$. Specifically,
when restricted to $\mathfrak{a}^{*}$,
$2\alpha^{\textsc{B}}_{i},2\alpha^{\textsc{F}}_{j}$,
$\alpha^{\textsc{B}}_{i}\pm\alpha^{\textsc{B}}_{i^{\prime}}$,
$\alpha^{\textsc{F}}_{j}\pm\alpha^{\textsc{F}}_{j^{\prime}}$, and
$\alpha^{\textsc{B}}_{i}\pm\alpha^{\textsc{F}}_{j}$ are positive restricted
roots in $\Sigma^{+}=\Sigma^{+}(\mathfrak{g},\mathfrak{a})$.

For the Harish-Chandra isomorphism $\gamma:=\gamma_{\mathfrak{h}}$ on the non-
diagonal $\mathfrak{h}$, we obtain it via the Cayley transform $c$. By Eq.
(19), $c$ sends the standard basis for $\mathfrak{t}$ the basis $\\{x_{\pm
i},y_{\pm j}\\}$ for $\mathfrak{h}$. Thus by a change of variable using $c$,
Theorem 2.3 and Eq. (8) imply

(36)
$\gamma:\mathfrak{Z}\rightarrow\mathfrak{P}(\mathfrak{h}^{*})=I_{\mathbb{C}}(x_{\pm
i},y_{\pm j})=\Lambda(\mathfrak{h}^{*}).$

The central character $\chi_{\lambda}$ for $\lambda\in\mathfrak{h}^{*}$ is now
given by

(37) $\chi_{\lambda}(z)=\gamma(z)(\lambda+\rho_{\mathfrak{h}}).$

For the Harish-Chandra isomorphism $\gamma^{\mathtt{0}}$, we simply apply Eqs.
(3, 4) to the choice of $\mathfrak{g},\mathfrak{k},\mathfrak{a}$ and
$\mathfrak{n}^{-}$ in this context. See Eqs. (3.2, 26) $\rho_{\mathfrak{h}}$
and $\rho$, the Weyl vectors of $\Sigma^{+}(\mathfrak{g},\mathfrak{h})$ and
$\Sigma^{+}$ respectively.

We also need the specialization of the spherical highest weights. Consider the
weight $\lambda^{\natural}_{\mathfrak{t}}$ (Eq. (28)) on $\mathfrak{t}$ and
$\mathfrak{b}^{\natural}$ (Eq. (33)). For $\lambda\in\mathscr{H}(p,q)$, we
define

$V(\lambda^{\natural}):=V(\lambda^{\natural}_{\mathfrak{t}},\mathfrak{b}^{\natural}).$

This is the $V_{\lambda}$ in the introduction. To detect the
$\mathfrak{k}$-sphericity (used in Section 4), we specialize Theorem 2.1,
which turns out to be quite simple (see [Zhu22, Proposition 4.4]).

######  Proposition 3.3.

Let $\lambda\in\mathscr{H}(p,q)$ and $V(\lambda^{\natural})$ be as above. If
$\lambda_{p}>\left\langle\lambda_{1}^{\prime}-p\right\rangle$, then $\dim
V(\lambda^{\natural})^{\mathfrak{k}}=1$.

###### Proof.

It is showed in [Zhu22, Theorem 4.3] that $V(\lambda^{\natural})$ is always
finite dimensional via some combinatorial machinery. This allows us to apply
the Cayley transform $c$ to rewrite $\mathfrak{t}$-weights as
$\mathfrak{h}$-weights by [Zhu22, Proposition 3.3]. The resulting weight is

(38)
$2\lambda_{\mathfrak{h}}^{\natural}:=\sum_{i=1}^{p}2\lambda_{i}\alpha_{i}^{\textsc{B}}+\sum_{j=1}^{q}2\left\langle\lambda_{j}^{\prime}-p\right\rangle\alpha_{j}^{\textsc{F}}\in\mathfrak{h}^{*}$

which vanishes on $\mathfrak{t}_{+}=\mathfrak{h}\cap\mathfrak{k}$. The high
enough condition degenerates to
$(2\lambda_{\mathfrak{h}}^{\natural}|_{\mathfrak{a}},\beta)>0$ for
$\beta=\alpha^{\textsc{B}}_{i}\pm\alpha^{\textsc{F}}_{j}$. Then a computation
shows the claim. ∎

Note that by definition of our notation, we have

(39)
$V(\lambda^{\natural})=V(\lambda^{\natural}_{\mathfrak{t}},\mathfrak{b}^{\natural})=V(2\lambda_{\mathfrak{h}}^{\natural},\mathfrak{b}).$

We write $2\overline{\lambda^{\natural}}$ for the restriction of
$2\lambda_{\mathfrak{h}}^{\natural}$ to $\mathfrak{a}$. Then by Theorem 2.2,
for spherical $V(\lambda^{\natural})$, $D_{\mu}\in\mathfrak{U}^{\mathfrak{k}}$
acts on $V(\lambda^{\natural})^{\mathfrak{k}}$ by the scalar
${\Gamma}(D_{\mu})(2\overline{\lambda^{\natural}}+\rho)$.

######  Remark 3.

For $p=q=1$, it is proved that every $\lambda\in\mathscr{H}(1,1)$ corresponds
to a spherical $V(\lambda^{\natural})$ [Zhu22, Theorem 1] utilizing Kac
induction which appears to be difficult for higher ranks.

##  4\. Harish-Chandra Isomorphisms and a Commutative Diagram

Let $\gamma$, ${\Gamma}$, and $\gamma^{\mathtt{0}}$ be as in Section 3. We
have $\operatorname{Im}\gamma=\Lambda(\mathfrak{h}^{*})$, the ring of
supersymmetric polynomials on $\mathfrak{h}^{*}$. The image of ${\Gamma}$
(equivalently of $\gamma^{\mathtt{0}}$), however, is less understood in such
terms. Alldridge’s work [All12] formulates
$\operatorname{Im}\gamma^{\mathtt{0}}$ as the intersection of some subalgebras
of $\mathfrak{S}(\mathfrak{a})$, while the second author proved a Weyl
groupoid invariance formulation in Type A in [Zhu22]. We will study the
$\operatorname{Im}\gamma^{\mathtt{0}}$ for $\mathfrak{g}=\mathfrak{gl}(2p|2q)$
and $\mathfrak{k}=\mathfrak{gl}(p|q)\oplus\mathfrak{gl}(p|q)$ in this section.
Specifically, we will give an independent proof that
$\ker{\Gamma}=(\mathfrak{U}\mathfrak{k})^{\mathfrak{k}}$ and
$\operatorname{Im}{\Gamma}=\operatorname{Im}\gamma^{\mathtt{0}}=\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$.

###  4.1. Image of $\gamma^{\mathtt{0}}$

We introduce even supersymmetric polynomials on par with what we did in
Subsection 2.2. Let $\\{x_{i}\\}:=\\{x_{i}\\}_{i=1}^{m}$ and
$\\{y_{j}\\}:=\\{y_{j}\\}_{j=1}^{n}$ be two sets of independent variables, and
set $\\{x_{i},y_{j}\\}:=\\{x_{i}\\}\cup\\{y_{j}\\}$. We write
$f(x_{1}=t,y_{1}=-t)$ for the polynomial obtained by the substitutions
$x_{1}=t$ and $y_{1}=-t$. A polynomial $f$ in $\\{x_{i},y_{j}\\}$ is said to
be even supersymmetric if (1) $f$ is invariant under permutations of
$\\{x_{i}\\}$ and of $\\{y_{j}\\}$ separately, and invariant under sign
changes of $\\{x_{i},y_{j}\\}$ and (2) $f(x_{1}=t,y_{1}=-t)$ is independent of
$t$. Denote the algebra of even supersymmetric polynomials in
$\\{x_{i},y_{j}\\}$ as $\mathscr{S}^{\mathtt{0}}_{m,n}$. It is not hard to see
that an even supersymmetric polynomial is supersymmetric in
$\\{x_{i}^{2},y_{j}^{2}\\}$, so $\mathscr{S}^{\mathtt{0}}_{m,n}$ is generated
by $p_{2r}(m,n)$ in Eq. (7) for $r\in\mathbb{Z}_{>0}$.

Let $\\{\epsilon_{i}\\}_{i=1}^{m}\cup\\{\delta_{j}\\}_{j=1}^{n}$ be the
standard basis for $V=\mathbb{C}^{m+n}$, equipped with an inner product
$(\cdot,\cdot)$ such that
$(\epsilon_{i},\epsilon_{j})=-(\delta_{i},\delta_{j})=c\delta_{i,j}$ and
$(\epsilon_{i},\delta_{j})=0$ for some $c>0$. Denote the coordinate functions
for this basis as $\\{x_{i}\\}$ and $\\{y_{j}\\}$. Then Condition (2) is
equivalent to (see [Mus12]) (2’) $f(X+\epsilon_{1}-\delta_{1})=f(X)$ if
$(X,\epsilon_{1}-\delta_{1})=0$. As such, $\mathscr{S}^{\mathtt{0}}_{m,n}$ is
identified as a subalgebra of $\mathfrak{P}(V^{*})$, and denoted as
$\Lambda^{\mathtt{0}}(V^{*})$ (c.f. $\Lambda(V^{*})$).

Let $W_{0}$ be the Weyl group associated with the restricted root system
$\Sigma:=\Sigma(\mathfrak{g},\mathfrak{a})$ (see Subsection 3.2). Then $W_{0}$
is in fact of Type BC and equals
$(\mathscr{S}_{p}\ltimes(\mathbb{Z}/2\mathbb{Z})^{p})\times(\mathscr{S}_{q}\ltimes(\mathbb{Z}/2\mathbb{Z})^{q})$.
Originally in [All12], $\operatorname{Im}{\Gamma}$ is given by the
intersection of certain subspaces using explicitly chosen elements. In
[Zhu22], it is proved that this description is in fact independent of any
choice, and we have

(40)
$\operatorname{Im}{\Gamma}=\Lambda^{\mathtt{0}}(\mathfrak{a}^{*}),\quad\mathfrak{U}^{\mathfrak{k}}/(\mathfrak{U}\mathfrak{k})^{\mathfrak{k}}\xrightarrow[\sim]{\gamma^{\mathtt{0}}}\Lambda^{\mathtt{0}}(\mathfrak{a}^{*}).$

We will offer a different proof that
$\ker{\Gamma}=(\mathfrak{U}\mathfrak{k})^{\mathfrak{k}}$ and
$\operatorname{Im}{\Gamma}=\operatorname{Im}\gamma^{\mathtt{0}}=\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$
below in Subsection 4.2.

######  Remark 4.

See [SV11, Zhu22] for an alternative way of writing down (even) supersymmetric
polynomials. Namely, we can consider a so-called Weyl groupoid that captures
both the usual double symmetry on two sets of coordinates, and the
translational invariance on hyperplanes for all isotropic (restricted) roots
(as in Condition (2’)). If $\mathfrak{W}$ (respectively $\mathfrak{W}_{0}$)
denotes the Weyl groupoid associated with $\Sigma(\mathfrak{g},\mathfrak{h})$
(respectively $\Sigma(\mathfrak{g},\mathfrak{a})$), then
$\operatorname{Im}\gamma=\mathfrak{P}(\mathfrak{h}^{*})^{\mathfrak{W}}$ while
$\operatorname{Im}\gamma^{\mathtt{0}}=\mathfrak{P}(\mathfrak{a}^{*})^{\mathfrak{W}_{0}}$.

In this section, we prove Theorem B. The strategy is to use a commutative
diagram (1) and the surjectivity of a certain restriction map we introduce
below. This map allows us to show that $\Lambda(\mathfrak{h}^{*})$ surjects
onto $\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$. Loosely speaking, the non-
restricted supersymmetry should cover the restricted supersymmetry entirely.

Recall $\mathfrak{h}=\mathfrak{t}_{+}\oplus\mathfrak{a}$. Let $p$ be the
respective projection from $\mathfrak{h}$ to $\mathfrak{a}$. Then we extend
$p$ to a projection homomorphism from $\mathfrak{S}(\mathfrak{h})$ to
$\mathfrak{S}(\mathfrak{a})$, again denoted as $p$. Under the identification
between $\mathfrak{S}(V)$ and $\mathfrak{P}(V^{*})$, we let
$\operatorname{\mathtt{Res}}$ be the “restriction” homomorphism corresponding
to $p$. Specifically, the adjoint map
$p^{*}:\mathfrak{a}^{*}\rightarrow\mathfrak{h}^{*}$ gives a pullback of a
polynomial $f$ defined on $\mathfrak{h}^{*}$ to a polynomial $g$ defined on
$\mathfrak{a}^{*}$, and $\operatorname{\mathtt{Res}}(f):=g$. We denote its
restriction on $\Lambda(\mathfrak{h}^{*})$ as $\operatorname{\mathtt{Res}}$
too.

######  Proposition 4.1.

The restriction map $\operatorname{\mathtt{Res}}$ on
$\Lambda(\mathfrak{h}^{*})$ surjects onto
$\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$.

###### Proof.

We show that $\operatorname{\mathtt{Res}}$ maps generators of
$\Lambda(\mathfrak{h}^{*})$ to all the generators of
$\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$. For this, we choose coordinate
systems for both $\mathfrak{P}(\mathfrak{h}^{*})$ and
$\mathfrak{P}(\mathfrak{a}^{*})$ as in Section 3. Throughout the proof we set
indices $i=1,\dots,p,j=1\dots,q$. Recall $x_{i}$, $y_{j}$, $x_{i}^{\prime}$,
$y_{j}^{\prime}$, $x_{\pm i}$, and $y_{\pm j}$ from Subsection 3.1. Also,
$x_{\pm i}$ and $y_{\pm j}$ span $\mathfrak{h}$ while $x_{i}$ and $y_{j}$ span
$\mathfrak{a}$ by Eq. (20). Thus by definition, the homomorphism
$\operatorname{\mathtt{Res}}$ on
$\mathfrak{P}(\mathfrak{h}^{*})=\mathbb{C}[x_{\pm i},y_{\pm j}]$ makes the
substitution

(41) $x_{\pm i}\mapsto\pm\frac{1}{2}x_{i},\quad y_{\pm
j}\mapsto\pm\frac{1}{2}y_{j}.$

The generators in Eq. (7) of $\Lambda(\mathfrak{h}^{*})$ become
$p_{r}^{(2p,2q)}(x_{\pm i},y_{\pm j})$. By Eq. (41), we have

$\operatorname{\mathtt{Res}}(p_{r}^{(2p,2q)}(x_{\pm i},y_{\pm
j}))=\begin{cases}2^{1-r}p_{2r}^{(p,q)}(x_{i},y_{j}),&\text{ if }r\text{ is
even }\\\ 0,&\text{ if }r\text{ is odd}\end{cases}.$

The preimage of $p_{2r}^{(p,q)}(x_{i},y_{j})$ under
$\operatorname{\mathtt{Res}}$ is $2^{r-1}p_{2r}^{(2p,2q)}(x_{\pm i},y_{\pm
j})$, meaning the generators of $\Lambda^{\mathtt{0}}$ are in the image of
$\operatorname{\mathtt{Res}}$. Therefore,
$\operatorname{Im}\operatorname{\mathtt{Res}}=\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$.
∎

The substitution in Eq. (41) also tells us that
$f\in\Lambda(\mathfrak{h}^{*})$ and
$\operatorname{\mathtt{Res}}(f)\in\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$ are
related by

(42)
$f\left(\sum_{i}a_{i}(\chi_{+i}-\chi_{-i})+\sum_{j}b_{j}(\eta_{+j}-\eta_{-j})\right)=(\operatorname{\mathtt{Res}}(f))\left(\sum_{i}2a_{i}\alpha_{i}^{\textsc{B}}+\sum_{j}2b_{j}\alpha_{j}^{\textsc{F}}\right).$

###  4.2. A commutative diagram

As pointed out in the Introduction, the center $\mathfrak{Z}$ acts on the
unique up to constant spherical vector by the character via $\gamma$.
Considered as a subalgebra of $\mathfrak{U}^{\mathfrak{k}}$, there is another
way to compute such a character using the Harish-Chandra map
$\gamma^{\mathtt{0}}$. We aim to relate the two Harish-Chandra isomorphisms
$\gamma$ and $\gamma^{\mathtt{0}}$.

We denote the quotient map
$\mathfrak{Z}\hookrightarrow\mathfrak{U}^{\mathfrak{k}}\rightarrow\mathfrak{U}^{\mathfrak{k}}/(\mathfrak{U}\mathfrak{k})^{\mathfrak{k}}$
as $\pi$. Note by definition,
$(\mathfrak{U}\mathfrak{k})^{\mathfrak{k}}\subseteq\mathfrak{U}\mathfrak{k}$
is contained in the kernel of the Harish-Chandra projection, hence in
$\ker{\Gamma}$. So the homomorphism
${\Gamma}^{\prime}:\mathfrak{D}=\mathfrak{U}^{\mathfrak{k}}/(\mathfrak{U}\mathfrak{k})^{\mathfrak{k}}\rightarrow\mathfrak{P}(\mathfrak{a}^{*})$
is well-defined. Then
$\operatorname{Im}{\Gamma}=\operatorname{Im}{\Gamma}^{\prime}$. Of course, the
bijectivity of ${\Gamma}^{\prime}$ would imply
${\Gamma}^{\prime}=\gamma^{\mathtt{0}}$. We form the following diagram first.

(43)
${\mathfrak{Z}}$${\mathfrak{D}}$${\Lambda(\mathfrak{h}^{*})}$${\mathfrak{P}(\mathfrak{a}^{*})}$$\scriptstyle{\pi}$$\scriptstyle{\gamma}$$\scriptstyle{\wr}$$\scriptstyle{{\Gamma}^{\prime}}$$\scriptstyle{\operatorname{\mathtt{Res}}}$

Then Theorem B follows from the two assertions below.

######  Proposition 4.2.

Diagram (43) is commutative.

######  Proposition 4.3.

The homomorphism ${\Gamma}^{\prime}$ is the isomorphism $\gamma^{\mathtt{0}}$
and
$\operatorname{Im}\gamma^{\mathtt{0}}=\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$.

###### Proof of Theorem B.

Combining Propositions 4.2 and 4.3, the above diagram (43) gives the
commutative diagram (see (1)):

${\mathfrak{Z}}$${\mathfrak{D}}$${\Lambda(\mathfrak{h}^{*})}$${\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})}$$\scriptstyle{\pi}$$\scriptstyle{\gamma}$$\scriptstyle{\wr}$$\scriptstyle{\gamma^{\mathtt{0}}}$$\scriptstyle{\wr}$$\scriptstyle{\operatorname{\mathtt{Res}}}$

By Proposition 4.1, $\operatorname{\mathtt{Res}}$ is surjective. Thus the
homomorphism $\pi$ is also surjective on $\mathfrak{D}$. ∎

Now we supply the proofs of Propositions 4.2 and 4.3.

###### Proof of Proposition 4.2.

The set of all the hook partitions $\lambda\in\mathscr{H}(p,q)$ satisfying the
condition in Proposition 3.3 give us a family of finite dimensional,
irreducible, and spherical $\mathfrak{g}$-module $V(\lambda^{\natural})$. In
notation of Eq. (22) (c.f. proof to Proposition 4.1), the corresponding
highest weight (see Eq. (38)) is:

(44)
$2\lambda^{\natural}_{\mathfrak{h}}=\sum_{i=1}^{p}\lambda_{i}(\chi_{+i}-\chi_{-i})+\sum_{j=1}^{q}\left\langle\lambda_{j}^{\prime}-p\right\rangle(\eta_{+j}-\eta_{-j})\in\mathfrak{h}^{*},$

whose restriction to $\mathfrak{a}$ is

(45)
$2\lambda^{\natural}_{\mathfrak{h}}|_{\mathfrak{a}}=2\overline{\lambda^{\natural}}=\sum_{i=1}^{p}2\lambda_{i}\alpha_{i}^{\textsc{B}}+\sum_{j=1}^{q}2\left\langle\lambda_{j}^{\prime}-p\right\rangle\alpha_{j}^{\textsc{F}}\in\mathfrak{a}^{*}.$

Let us denote the set of weights in Eq. (45) as $\mathbb{K}$. Now, considered
as a subset of the integral lattice $(2\mathbb{Z})^{p+q}$ in
$\mathfrak{a}^{*}\cong\mathbb{C}^{p+q}$, $\mathbb{K}$ is fully determined by
$p+q$ many inequalities

$x_{1}\geq\cdots\geq x_{p}>y_{1}\geq\cdots\geq y_{q}\geq 0.$

Then $\mathbb{K}$ is Zariski dense in $\mathfrak{a}^{*}$ as it is the
intersection of an open cone with $(2\mathbb{Z})^{p+q}$.

For $z\in\mathfrak{Z}\subseteq\mathfrak{U}^{\mathfrak{k}}$, it acts by a
scalar by Lemma 2.1, and by Theorem 2.2, this scalar is

${\Gamma}(z)(2\overline{\lambda^{\natural}}+\rho)={\Gamma}^{\prime}(\pi(z))(2\overline{\lambda^{\natural}}+\rho).$

On the other hand, by Eq. (37), $z$ acts on the entirety of
$V(\lambda^{\natural})$ by
$\gamma(z)(2\lambda^{\natural}_{\mathfrak{h}}+\rho_{\mathfrak{h}})$. Hence

(46)
$\gamma(z)(2\lambda^{\natural}_{\mathfrak{h}}+\rho_{\mathfrak{h}})={\Gamma}^{\prime}(\pi(z))(2\overline{\lambda^{\natural}}+\rho).$

But $2\lambda^{\natural}_{\mathfrak{h}}+\rho_{\mathfrak{h}}$ vanishes on
$\mathfrak{t}_{+}$. Therefore, the left side of Eq. (46) is just
$\operatorname{\mathtt{Res}}(\gamma(z))(2\overline{\lambda^{\natural}}+\rho)$.
This can be verified directly using Eq. (42) with inputs Eqs. (3.2, 26, 44).
Thus Eq. (46) implies

$\operatorname{\mathtt{Res}}(\gamma(z))={\Gamma}^{\prime}(\pi(z))$

on $\mathbb{K}+\rho$. As the two sides agree on a Zariski dense subset of
$\mathfrak{a}^{*}$, they must be equal everywhere. This proves the
commutativity of (43). ∎

###### Proof of Proposition 4.3.

Now we show that
$\operatorname{Im}{\Gamma}^{\prime}=\operatorname{Im}{\Gamma}=\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$,
and ${\Gamma}^{\prime}$ is injective. For the sake of simplicity, write
$\mathscr{H}$ for $\mathscr{H}(p,q)$, and let
$\mathscr{H}_{d}=\\{\lambda\in\mathscr{H}:|\lambda|\leq d\\}$. We also let
$\mathfrak{P}_{2d}(\mathfrak{a}^{*}):=\\{f\in\mathfrak{P}(\mathfrak{a}^{*}):\deg
f\leq 2d\\}$, and
$\Lambda^{\mathtt{0}}_{d}(\mathfrak{a}^{*}):=\\{f\in\Lambda^{\mathtt{0}}(\mathfrak{a}^{*}):\deg
f\leq 2d\\}$.

Step 1. Let
$\mathfrak{P}:=\mathfrak{U}(\mathfrak{p})=\mathfrak{S}(\mathfrak{p}^{-})\mathfrak{S}(\mathfrak{p}^{+})$.
By the Poincaré–Birkhoff–Witt Theorem, we have
$\mathfrak{U}=\mathfrak{U}\mathfrak{k}\oplus\mathfrak{P}$. This is in fact a
$\mathfrak{k}$-module decomposition, and its $\mathfrak{k}$-invariant
counterpart is

$\mathfrak{U}^{\mathfrak{k}}=(\mathfrak{U}\mathfrak{k})^{\mathfrak{k}}\oplus\mathfrak{P}^{\mathfrak{k}}.$

By Definition 3.2, we see that
$D_{\lambda}\in(W_{\lambda}^{*}W_{\lambda})^{\mathfrak{k}}$ constitute a basis
for

(47)
$\mathfrak{P}^{\mathfrak{k}}=\bigoplus_{\lambda\in\mathscr{H}}(W_{\lambda}^{*}W_{\lambda})^{\mathfrak{k}}$

indexed by $\lambda\in\mathscr{H}$. Let
$\mathscr{D}_{\lambda}\in\mathfrak{U}^{\mathfrak{k}}/(\mathfrak{U}\mathfrak{k})^{\mathfrak{k}}=\mathfrak{D}$
be the equivalent class of $D_{\lambda}\in\mathfrak{U}^{\mathfrak{k}}$.
Identifying
$\mathfrak{P}^{\mathfrak{k}}\cong\mathfrak{U}^{\mathfrak{k}}/(\mathfrak{U}\mathfrak{k})^{\mathfrak{k}}$,
we see that $\mathscr{D}_{\lambda}$ gives a basis for $\mathfrak{D}$.

Step 2. Furthermore, we see that $\mathfrak{P}^{\mathfrak{k}}=\bigoplus_{d\geq
0}(\mathfrak{S}^{d}(\mathfrak{p}^{-})\mathfrak{S}^{d}(\mathfrak{p}^{+}))^{\mathfrak{k}}$
is graded by non-negative integers according to $|\lambda|$ in Eq. (47) (c.f.
Eq. (27)) as a vector superspace. The basis $\\{\mathscr{D}_{\lambda}\\}$ is
homogeneous. We further define

$\mathfrak{D}_{d}:=\operatorname{Span}\\{\mathscr{D}_{\lambda}:\lambda\in\mathscr{H}_{d}\\}.$

This gives a vector superspace filtration. The reason why we consider
filtration instead of grading is that ${\Gamma}^{\prime}$ only preserves
filtration due to the $\rho$ shift in its definition. For each filtered
degree, let

${\Gamma}^{\prime}_{d}:\mathfrak{D}_{d}\rightarrow\mathfrak{P}_{2d}(\mathfrak{a}^{*}).$

Step 3. By Proposition 4.1, $\operatorname{\mathtt{Res}}$ is surjective onto
$\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$, from which we have

$\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})=\operatorname{\mathtt{Res}}(\gamma(\mathfrak{Z}))$

We also showed that the diagram commutes. Thus the right side is
${\Gamma}^{\prime}(\pi(\mathfrak{Z}))$, which gives
$\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})\subseteq\operatorname{Im}{\Gamma}^{\prime}$.
By Step 2., the filtered version of the assertion is

(48)
$\Lambda^{\mathtt{0}}_{d}(\mathfrak{a}^{*})\subseteq\operatorname{Im}{\Gamma}^{\prime}_{d}.$

Step 4. By Proposition 6.1 below, the polynomials $J_{\lambda}$ with
$|\lambda|\leq d$ give a basis for
$\Lambda^{\mathtt{0}}_{d}(\mathfrak{a}^{*})$. Such $J_{\lambda}$ are indexed
again by $\lambda\in\mathscr{H}_{d}$. Therefore,
$\dim\Lambda^{\mathtt{0}}_{d}(\mathfrak{a}^{*})=|\mathscr{H}_{d}|\geq\dim\operatorname{Im}{\Gamma}^{\prime}_{d}$,
proving

$\Lambda^{\mathtt{0}}_{d}(\mathfrak{a}^{*})=\operatorname{Im}{\Gamma}^{\prime}_{d}$

by Eq. (48). This shows
$\operatorname{Im}{\Gamma}^{\prime}=\operatorname{Im}{\Gamma}=\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$,
as well as the injectivity of ${\Gamma}^{\prime}$. Therefore,
${\Gamma}^{\prime}=\gamma^{\mathtt{0}}$ is bijective, from which we have
$\gamma^{\mathtt{0}}:\mathfrak{D}\overset{\sim}{\rightarrow}\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$.
∎

##  5\. Generalized Verma Modules

In this section, we study certain generalized Verma modules, denoted as
$I_{\lambda}$. These are the $\mathfrak{g}$-modules induced on $W_{\lambda}$
($\lambda\in\mathscr{H}(p,q)$) from a parabolic subalgebra containing
$\mathfrak{k}$ to $\mathfrak{g}$. If we choose a Borel subalgebra of
$\mathfrak{g}$ as the parabolic subalgebra, then we obtain the usual Verma
module, which explains the name. We show that $I_{\lambda}$ is spherical. We
investigate a natural grading on $I_{\lambda}$, and use it to show that
certain central elements $Z_{\mu}$ act trivially on $I_{\lambda}$ trivially
whenever $|\lambda|\leq|\mu|$. Further strengthening is also obtained,
asserting the same vanishing action of $Z_{\mu}$ for $\lambda\nsupseteq\mu$.
This is Theorem C.

###  5.1. Basics

Consider the parabolic subalgebra
$\mathfrak{q}:=\mathfrak{k}\oplus\mathfrak{p}^{+}$ in $\mathfrak{g}$. The
associated set of roots is given by
$\Sigma(\mathfrak{k},\mathfrak{t})\cup\Sigma(\mathfrak{p}^{+},\mathfrak{t})$.
Let $W_{\lambda}$ be as in the Cheng–Wang decomposition. We extend the
$\mathfrak{k}$-action trivially to $\mathfrak{p}^{+}$ to obtain a
$\mathfrak{q}$-module structure on $W_{\lambda}$. We define the generalized
Verma module as

$I_{\lambda}:=\operatorname{Ind}_{\mathfrak{k}+\mathfrak{p}^{+}}^{\mathfrak{g}}W_{\lambda}=\mathfrak{U}\otimes_{\mathfrak{U}(\mathfrak{q})}W_{\lambda}.$

By the Poincaré–Birkhoff–Witt theorem, we see that
$\mathfrak{U}=\mathfrak{S}(\mathfrak{p}^{-})\mathfrak{U}(\mathfrak{k})\mathfrak{S}(\mathfrak{p}^{+})$,
and we have

$I_{\lambda}\cong\mathfrak{S}(\mathfrak{p}^{-})\otimes W_{\lambda}$

both as super vector spaces and as $\mathfrak{k}$-modules. Clearly,
$I_{\lambda}$ is a weight module and the highest weight is Eq. (28), given
with respect to $\mathfrak{b}^{\natural}$ (see Eq. (33)). Then
$V(\lambda^{\natural})$ is the irreducible quotient of $I_{\lambda}$.

Next, we introduce a grading on $I_{\lambda}$. Recall that in the one-
dimensional center of $\mathfrak{k}$, there is a special element $J$ (see
Section 3) such that the Harish-Chandra decomposition
$\mathfrak{g}=\mathfrak{p}^{-}\oplus\mathfrak{k}\oplus\mathfrak{p}^{+}$
corresponds to $(-1,0,1)$-eigenspaces of $\operatorname{ad}J$, and gives rise
to the short grading. We may extend the action of $\operatorname{ad}J$ to
$\mathfrak{U}=\mathfrak{S}(\mathfrak{p}^{-})\mathfrak{U}(\mathfrak{k})\mathfrak{S}(\mathfrak{p}^{+})$.
Let $\\{\xi_{i}\\},\\{x_{i}\\}$ and $\\{\eta_{i}\\}$ be homogeneous bases for
$\mathfrak{p}^{-}$, $\mathfrak{k}$, and $\mathfrak{p}^{+}$ respectively. Then
the Poincaré–Birkhoff–Witt basis is:

$\left\\{\xi_{i_{1}}\cdots\xi_{i_{m}}x_{j_{1}}\cdots
x_{j_{l}}\eta_{k_{1}}\cdots\eta_{k_{n}}:i_{1}\leq\cdots\leq
i_{m},j_{1}\leq\cdots\leq j_{l},k_{1}\leq\cdots\leq k_{n}\right\\}.$

where a strict inequality of subindices occurs when the corresponding basis
vectors are odd. We let

$\mathfrak{U}^{d}:=\operatorname{Span}\\{\xi_{i_{1}}\cdots\xi_{i_{m}}x_{j_{1}}\cdots
x_{j_{l}}\eta_{k_{1}}\cdots\eta_{k_{n}}:n-m=d\\}.$

######  Lemma 5.1.

The universal enveloping algebra $\mathfrak{U}$ is graded by
$\operatorname{ad}J$, and
$\mathfrak{U}=\bigoplus_{d\in\mathbb{Z}}\mathfrak{U}^{d}$.

###### Proof.

The action of $J$ on the basis $\xi_{i_{1}}\cdots\xi_{i_{m}}x_{j_{1}}\cdots
x_{j_{l}}\eta_{k_{1}}\cdots\eta_{k_{n}}$ is given by the scalar $n-m=d$ in
$\mathfrak{U}^{d}$. The bases for $\mathfrak{U}^{d}$ for all $d\geq 0$ give a
partition of the Poincaré–Birkhoff–Witt basis, and the sum is hence direct.
Furthermore, the multiplication in $\mathfrak{U}$ respects the Lie
superbracket, which in turn respects the short grading on $\mathfrak{g}$.
Therefore the grading is well-defined. ∎

Let $l=|\lambda|$. Then $I_{\lambda}$ is also endowed with the
$\operatorname{ad}J$ grading. Explicitly,

$I_{\lambda}^{l-d}:=\mathfrak{S}^{d}(\mathfrak{p}^{-})\otimes W_{\lambda}$

has degree $l-d$ for $d\in\mathbb{N}$. In particular, its top homogeneous
component is $\mathbb{C}\otimes W_{\lambda}$ of degree $l$, isomorphic to
$W_{\lambda}$. By Lemma 5.1, since the $\mathfrak{g}$ acts by left
multiplication on $I_{\lambda}$, we have
$\mathfrak{U}^{d}.I_{\lambda}^{m}\subseteq I_{\lambda}^{d+m}$. This leads to
the following lemma.

######  Lemma 5.2.

Let $u\in\mathfrak{S}^{k}(\mathfrak{p}^{+})$. Then $u.I_{\lambda}^{m}\subseteq
I_{\lambda}^{k+m}$.

Next, we show that $I_{\lambda}$ is in fact spherical.

######  Proposition 5.3.

The module $I_{\lambda}$ is spherical. Furthermore, $\dim
I_{\lambda}^{\mathfrak{k}}=1$ and $I_{\lambda}^{\mathfrak{k}}\subseteq
I_{\lambda}^{0}$.

###### Proof.

As a $\mathfrak{k}$-module, we have
$I_{\lambda}\cong\bigoplus_{\nu\in\mathscr{H}(p,q)}W_{\nu}^{*}\otimes
W_{\lambda}$ and each component can be identified as
$\operatorname{Hom}(W_{\nu},W_{\lambda})$. Only the degree 0 component
$W_{\lambda}^{*}\otimes
W_{\lambda}\cong\operatorname{End}\left(W_{\lambda}\right)$ has a one-
dimensional $\mathfrak{k}$-invariant subspace by Schur’s Lemma for Type M
modules. Hence $I_{\lambda}$ is spherical with a unique up to constant
spherical vector. ∎

By the above Proposition, $\omega\neq 0$ in $I_{\lambda}^{\mathfrak{k}}$ is
unique up to constant. Let us fix such $\omega\in I_{\lambda}^{\mathfrak{k}}$
in the following discussion.

###  5.2. Vanishing actions

Recall that the supersymmetric Shimura operator $D_{\mu}$ in
$\mathfrak{U}^{\mathfrak{k}}$ is defined as the image of $1_{\mu}$ under the
following composition of maps (Definition 3.2)

$\displaystyle\left(W_{\mu}^{*}\right.\otimes\left.W_{\mu}\right)^{\mathfrak{k}}\hookrightarrow\left(\mathfrak{S}(\mathfrak{p}^{-})\otimes\mathfrak{S}(\mathfrak{p}^{+})\right)^{\mathfrak{k}}\rightarrow$
$\displaystyle\mathfrak{U}^{\mathfrak{k}}$ $\displaystyle
1_{\mu}\xmapsto{\hphantom{\left(W_{\mu}^{*}\right.\otimes\left.W_{\mu}\right)^{\mathfrak{k}}\hookrightarrow\left(\mathfrak{S}(\mathfrak{p}^{-})\otimes\mathfrak{S}(\mathfrak{p}^{+})\right)^{\mathfrak{k}}\rightarrow\mathfrak{U}^{\mathfrak{k}}}}$
$\displaystyle D_{\mu}$

In particular $D_{\mu}\in W_{\mu}^{*}W_{\mu}$. If we take a homogeneous basis
$\\{\eta^{\mu}_{i}\\}$ for $W_{\mu}$ and its dual basis $\\{\xi^{-\mu}_{i}\\}$
for $W_{\mu}^{*}$, then $D_{\mu}$ can be written as

(49) $D_{\mu}=\sum\xi^{-\mu}_{i}\eta^{\mu}_{i}.$

Let $m=|\mu|$ and $l=|\lambda|$. We have the following “vanishing” actions
(Propositions 5.4 and 5.6).

######  Proposition 5.4.

If $l<m$, then $D_{\mu}.v^{0}=0$ for any $v^{0}\in I_{\lambda}^{0}$.

###### Proof.

In $D_{\mu}$, every $\eta^{\mu}_{i}$ in Eq. (49) has degree $m$ as an element
in $\mathfrak{S}^{m}(\mathfrak{p}^{+})$. Hence
$\eta^{\mu}_{i}.I_{\lambda}^{0}\subseteq I_{\lambda}^{m}$ by Lemma 5.2. The
top degree of $I_{\lambda}$ is $l$. But $l<m$. Thus
$D_{\mu}.I_{\lambda}^{0}=I_{\lambda}^{m}=\\{0\\}$. ∎

Viewing $W_{\mu}\subseteq\mathfrak{U}$, we have the representation
$R:W_{\mu}\otimes I_{\lambda}\rightarrow I_{\lambda}$ given by $\eta\otimes
v\mapsto\eta.v$ for $\eta\in W_{\mu}$ and $v\in I_{\lambda}$.

######  Lemma 5.5.

The image of $R$ restricted to $W_{\mu}\otimes I_{\lambda}^{\mathfrak{k}}$ is
homomorphic to $W_{\mu}$.

###### Proof.

Let $m=|\mu|$. For each $\eta^{\mu}_{i}.\omega\in I_{\lambda}^{m}$, if we
apply a homogeneous $X\in\mathfrak{k}$, we get

(50)
$X.(\eta^{\mu}_{i}.\omega)=[X,\eta^{\mu}_{i}].\omega+(-1)^{|X||\eta^{\mu}_{i}|}\eta^{\mu}_{i}.X.\omega=(\operatorname{ad}(X)(\eta^{\mu}_{i})).\omega$

as $X.\omega=0$. Thus, the map $\eta^{\mu}_{i}\mapsto\eta^{\mu}_{i}.\omega$
intertwines with the actions of $\mathfrak{k}$ on $W_{\mu}$ and
$I_{\lambda}^{\mathfrak{k}}$.

Let $S$ be the subspace of $I_{\lambda}^{m}$ spanned by
$\\{\eta^{\mu}_{i}.\omega\\}$. Then $S$ is a homomorphic image of $W_{\mu}$ in
$I_{\lambda}$ by Eq. (50). ∎

######  Proposition 5.6.

If $l=m$ but $\lambda\neq\mu$, then $D_{\mu}.\omega=0$.

###### Proof.

Since $W_{\mu}\subseteq\mathfrak{S}^{l}(\mathfrak{p}^{+})$ and
$I_{\lambda}^{\mathfrak{k}}\subseteq I_{\lambda}^{0}$, by Lemma 5.2, we see
that

$W_{\mu}\otimes I_{\lambda}^{\mathfrak{k}}\rightarrow
I_{\lambda}^{l}=\mathbb{C}\otimes W_{\lambda}.$

By Lemma 5.5, $W_{\mu}$ is homomorphic to some $S$ in $I_{\lambda}^{l}$. Since
$\lambda\neq\mu$, $I_{\lambda}^{l}=\\{0\\}$. Hence, $W_{\mu}$ acts on
$I_{\lambda}^{\mathfrak{k}}$ by 0, and so does $D_{\mu}$. ∎

######  Corollary 5.7.

For any $\lambda\in\mathscr{H}(p,q)$ such that $|\lambda|\leq|\mu|$,
$\lambda\neq\mu$, we have $D_{\mu}.\omega=0$ in $I_{\lambda}$.

###### Proof.

It follows from Propositions 5.3, 5.4 for the case $|\lambda|<|\mu|$, and 5.6
for the case $|\lambda|=|\mu|$. ∎

Recall $\pi(D_{\mu})=\mathscr{D}_{\mu}$. By Theorem B, $\pi$ is surjective.
Then it is possible to find $Z_{\mu}\in\mathfrak{Z}$ such that
$\pi(Z_{\mu})=\mathscr{D}_{\mu}$. We show that $Z_{\mu}$ acts by 0 on
$I_{\lambda}$ when $\lambda$ satisfies certain condition.

######  Proposition 5.8.

The central element $Z_{\mu}$ acts on $I_{\lambda}$ by $0$ when
$|\lambda|\leq|\mu|$, and $\lambda\neq\mu$.

###### Proof.

By Lemma 2.1, $D_{\mu}$ acts on $\omega\in I_{\lambda}$ by a scalar
$c_{\mu}(\lambda)$ for any $\lambda$. On the other hand, the action of
$\mathfrak{U}^{\mathfrak{k}}$ on a spherical vector $v^{\mathfrak{k}}$ in a
$\mathfrak{g}$-module descends to an action of
$\mathfrak{D}=\mathfrak{U}^{\mathfrak{k}}/(\mathfrak{U}\mathfrak{k})^{\mathfrak{k}}$
by setting $\pi(X).v^{\mathfrak{k}}:=X.v^{\mathfrak{k}}$. Hence, in
$I_{\lambda}$, there is no ambiguity of writing

(51)
$c_{\mu}(\lambda)\omega=D_{\mu}.\omega=\mathscr{D}_{\mu}.\omega=Z_{\mu}.\omega.$

Therefore, by Corollary 5.7,

(52) $c_{\mu}(\lambda)=0,\;\text{if }|\lambda|\leq|\mu|,\lambda\neq\mu.$

Since $Z_{\mu}\in\mathfrak{Z}$ acts on the entirety of a highest weight module
by a scalar and $I_{\lambda}$ is a highest weight module, such scalar has to
be $c_{\mu}(\lambda)$ which vanishes according to Eq. (52). ∎

###  5.3. A stronger result

What is proved in the previous subsection turns out to be enough for the
purposes of this paper (Theorem A). Nonetheless, we give a better description
of such vanishing actions, proving Theorem C.

First, we recall the following “weight decomposition” proposition from
[Zhu22], parallel to Proposition 3.2 in [Kum10], and is first due to Kostant
[Kos59]. We let $\mathfrak{L}$ be a Lie (super)algebra,
$\mathfrak{b}\subseteq\mathfrak{L}$ be a Borel subalgebra.

######  Proposition 5.9.

Let $L_{1}$ and $L_{2}$ be two finite dimensional irreducible
$\mathfrak{L}$-modules with highest weight $\kappa_{1}$ and $\kappa_{2}$, and
$V$ be a weight module of $\mathfrak{L}$. If
$\operatorname{Hom}_{\mathfrak{L}}\left(L_{1},L_{2}\otimes
V\right)\neq\\{0\\}$, then $\kappa_{1}=\kappa_{2}+\nu$ for some weight $\nu$
of $V$.

We would like to show the following branching statement, i.e., $W_{\mu}$
occurs in $I_{\lambda}$ as a homomorphic image if $\mu_{i}\leq\lambda_{i}$.
This result is similar to the one regarding the irreducible representation
$V(\lambda^{\natural})$ in [Zhu22].

######  Proposition 5.10.

If
$\operatorname{Hom}_{\mathfrak{k}}\left(W_{\mu},I_{\lambda}\right)\neq\\{0\\}$,
then $\lambda\supseteq\mu$.

###### Proof.

The $\mathfrak{k}$-module structure on $I_{\lambda}$ is given by
$\mathfrak{S}(\mathfrak{p}^{-})\otimes W_{\lambda}$. Let us denote the
representation map on $\mathfrak{S}(\mathfrak{p}^{-})\otimes W_{\lambda}$
(respectively $W_{\lambda}\otimes\mathfrak{S}(\mathfrak{p}^{-})$) as $\pi$
(respectively $\pi^{\prime}$). Then the braiding map
$s:\mathfrak{S}(\mathfrak{p}^{-})\otimes W_{\lambda}\rightarrow
W_{\lambda}\otimes\mathfrak{S}(\mathfrak{p}^{-})$ gives a canonical module
isomorphism, meaning that if $W_{\mu}$ occurs in $I_{\lambda}$, then it also
occurs in $W_{\lambda}\otimes\mathfrak{S}(\mathfrak{p}^{-})$.

By Proposition 5.9,
$\nu=\mu^{\natural}_{\mathfrak{t}}-\lambda^{\natural}_{\mathfrak{t}}$ is a
weight of $\mathfrak{S}(\mathfrak{p}^{-})$. Here,
$\mu^{\natural}_{\mathfrak{t}}$ and $\lambda^{\natural}_{\mathfrak{t}}$ are
weights in $\mathfrak{t}^{*}$. As both $\mu^{\natural}_{\mathfrak{t}}$ and
$\lambda^{\natural}_{\mathfrak{t}}$ are integral combinations of
$\gamma\in\Sigma_{\perp}$, so is $\nu$. Since
$\Sigma_{\perp}\subseteq\Sigma(\mathfrak{p}^{+}_{\overline{0}},\mathfrak{t})$,
such integral coefficients in $\nu$ must be non-positive. Equivalently,
$\lambda\supseteq\mu$. ∎

###### Proof of Theorem C.

As in the proof of Proposition 5.8, $c_{\mu}(\lambda)$ is the scalar by which
$D_{\mu}$ acts on $I_{\lambda}^{\mathfrak{k}}$. Again by Eq. (51) and the fact
that $Z_{\mu}$ has to act uniformly by a scalar on $I_{\lambda}$,
$c_{\mu}(\lambda)$ is the scalar by which $D_{\mu}$ acts on the entirety of
$I_{\lambda}$. By Lemma 5.5, the representation of $W_{\mu}$ on
$I_{\lambda}^{\mathfrak{k}}$ always gives a homomorphic image of itself in
$I_{\lambda}$. If $\lambda\nsupseteq\mu$, then this image is zero by
Proposition 5.10. Spelling out $D_{\mu}$ as in Eq. (49), we see that
$D_{\mu}.I_{\lambda}^{\mathfrak{k}}\subseteq
W_{\mu}.I_{\lambda}^{\mathfrak{k}}=\\{0\\}$. Thus,

(53) $c_{\mu}(\lambda)=0,\;\text{if }\lambda\nsupseteq\mu,$

and this proves that $Z_{\mu}$ acts on $I_{\lambda}$ by 0 unless
$\lambda_{i}\geq\mu_{i}$ for all $i$. ∎

Eq. (53) gives more vanishing points of $c_{\mu}$ than we actually need for
proving Theorem A in Section 6. It is stronger than Proposition 5.8 which
concerns only zeros at lower degrees. In fact, this is much closer to the
original formulation of the vanishing properties, introduced in Theorem 2.5,
c.f. Proposition 6.1 below.

##  6\. Type BC Interpolation Polynomials

In this section, we specify Sergeev and Veselov’s results (Subsection 2.4)
with all the parameters in Table 1 (Subsection 3.2), and then an explicit
change of variables. For the sake of simplicity, write $\mathscr{H}$ for
$\mathscr{H}(p,q)$, $\mathscr{H}_{d}$ for
$\\{\lambda\in\mathscr{H}:|\lambda|\leq d\\}$. Throughout the section, we set
$1\leq i\leq p$, $1\leq j\leq q$.

We now specify using our restricted root system
$\Sigma=\Sigma(\mathfrak{g},\mathfrak{a})$. In particular, we set $m:=q,n:=p$,
$\epsilon_{j}:=\alpha^{\textsc{F}}_{j}$,
$\delta_{i}:=\alpha^{\textsc{B}}_{i}$, $z_{i}:=x_{i}$, $w_{j}:=y_{j}$. By
Table 1, $\mathsf{k}=-1$, $\mathsf{q}=\mathsf{s}=-1/2$ while $\mathsf{p}$ and
$\mathsf{r}$ do not exist. By comparing Eq. (10) with $(-1/2)\rho$ as in Eq.
(26), we have $\mathsf{h}=q-p+1/2$. Thus we have the two parameters

$\mathsf{k}=-1,\quad\mathsf{h}=q-p+1/2$

for $\Lambda^{(-1,q-p+1/2)}_{q,p}$. Note our form Eq. (21) is the negative of
what is defined in Subsection 2.4. But this does not matter when
$\mathsf{k}=-1$. We set (c.f. Eq. (9))

$\overline{\lambda^{\natural}}:=\sum\lambda_{i}\alpha^{\textsc{B}}_{i}+\sum\langle\lambda_{j}^{\prime}-m\rangle\alpha^{\textsc{F}}_{j}.$

We write $\Lambda^{\mathtt{0}}$ for $\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$,
and define $\Lambda^{\mathtt{0}}_{d}:=\\{f\in\Lambda^{\mathtt{0}}:\deg f\leq
2d\\}$. We define
$2(\mathscr{H}_{d})^{\natural}+\rho:=\left\\{2\overline{\lambda^{\natural}}+\rho:\lambda\in\mathscr{H}_{d}\right\\}\subseteq\mathfrak{a}^{*}$.

######  Proposition 6.1.

For each $\mu\in\mathscr{H}$, there is a unique polynomial
$J_{\mu}\in\Lambda^{\mathtt{0}}$ of degree $2|\mu|$ such that

$J_{\mu}(2\overline{\lambda^{\natural}}+\rho)=0,\quad\text{for all
}\lambda\nsupseteq\mu,\,\lambda\in\mathscr{H}$

and that
$J_{\mu}(2\overline{\mu^{\natural}}+\rho)=C_{\mu}^{-}(1;-1)C_{\mu}^{+}(2q-2p;-1)$.
Moreover, $\\{J_{\mu}:\mu\in\mathscr{H}_{d}\\}$ is a basis for
$\Lambda^{\mathtt{0}}_{d}$.

###### Proof.

Let $\tau:x_{i}\mapsto\frac{1}{2}(x_{i}-\rho_{i}^{\textsc{B}})$ and
$y_{j}\mapsto\frac{1}{2}(y_{j}-\rho_{j}^{\textsc{F}})$ be the change of
variables which is an isomorphism from $\Lambda^{(-1,q-p+1/2)}_{p,q}$ to
$\Lambda^{\mathtt{0}}(\mathfrak{a}^{*})$ (c.f. [Zhu22, Proposition 5.9]).
Consider
$I_{\mu^{\prime}}(X;-1,q-p+\frac{1}{2})\in\Lambda^{(-1,q-p+1/2)}_{p,q}$. Since
$(z(\lambda),w(\lambda))=(\lambda^{\prime})^{\natural}$, by Theorem 2.5, we
see that $J_{\mu}:=\tau(I_{\mu^{\prime}}(X;-1,q-p+\frac{1}{2}))$ satisfies the
vanishing properties

(54) $J_{\mu}(2\overline{\lambda^{\natural}}+\rho)=0,\quad\text{for all
}\lambda\nsupseteq\mu,\,\lambda\in\mathscr{H}.$

The normalization condition is straightforward. As
$\\{I_{\mu}^{\prime}:\mu\in\mathscr{H}_{d}\\}$ is a basis for
$\\{f\in\Lambda^{(-1,q-p+1/2)}_{p,q}:\deg f\leq 2d\\}$,
$\\{J_{\mu}:\mu\in\mathscr{H}_{d}\\}$ becomes a basis for
$\Lambda^{\mathtt{0}}_{d}$. ∎

######  Proposition 6.2.

Every $f\in\Lambda^{\mathtt{0}}_{d}$ is determined by its values on
$2(\mathscr{H}_{d})^{\natural}+\rho$.

###### Proof.

Let $\mathcal{V}_{d}$ be the space of functions on
$2(\mathscr{H}_{d})^{\natural}+\rho$. Then
$\dim\Lambda^{\mathtt{0}}_{d}=\dim\mathcal{V}_{d}=|\mathscr{H}_{d}|$. In
particular, $\mathcal{V}_{d}$ has a Kronecker-delta basis
$\\{\delta_{\lambda}:\delta_{\lambda}(2\overline{\lambda^{\natural}}+\rho)=1,\delta_{\lambda}(2\overline{\mu^{\natural}}+\rho)=0,\lambda,\mu\in\mathscr{H}_{d}\\}$.
Next, the evaluation of $f\in\Lambda^{\mathtt{0}}_{d}$ on $\mathscr{H}_{d}$
gives a restriction map

$\mathtt{res}:\Lambda^{\mathtt{0}}_{d}\rightarrow\mathcal{V}_{d}.$

To prove the statement, we show that $\mathtt{res}$ is an isomorphism.

Fix a total order $\succ$ on $\mathscr{H}_{d}$ such that $\mu\succ\lambda$
implies $|\mu|\geq|\lambda|$. Consider the matrix $R$ for $\mathtt{res}$ with
respect to the bases $\\{J_{\mu}\\}$ for $\Lambda^{\mathtt{0}}_{d}$ and
$\\{\delta_{\lambda}\\}$ for $\mathcal{V}_{d}$ arranged by $\succ$. Since
$J_{\mu}(2\overline{\mu^{\natural}}+\rho)\neq 0$, and
$J_{\mu}(2\overline{\lambda^{\natural}}+\rho)=0$ for any $\lambda$ such that
$\mu\succ\lambda$, we see that $R$ is upper triangular with non-zero diagonal
entries. Therefore $R$ is invertible, proving the statement. ∎

To prove Theorem A, we first show that
$\gamma^{\mathtt{0}}(\mathscr{D}_{\mu})$ is proportional to $J_{\mu}$ defined
above. Then we pin down the scalar

(55) $k_{\mu}:=(-1)^{|\mu|}C_{\mu}^{-}(1;-1).$

by comparing the top homogeneous degrees of both sides. Here

$C_{\mu}^{-}(1;-1)=\prod_{(i,j)\in\mu}\left(\mu_{i}-j+\mu^{\prime}_{j}-i+1\right)$

(c.f. Subsection 2.4). The factor of $(-1)^{|\mu|}$ is due to the definition
of $d_{\mu}$, which is a result of the definition of $\varphi_{T}$ in [SV05,
Eq. (41)].

###### Proof of Theorem A.

We show that

$\gamma^{\mathtt{0}}(\mathscr{D}_{\mu})=k_{\mu}J_{\mu}.$

Step 1. From the definition of ${\Gamma}$ and $\gamma^{\mathtt{0}}$, we see
that $\deg\gamma^{\mathtt{0}}(\mathscr{D}_{\mu})\leq 2|\mu|$. By (1), we have
$\gamma^{\mathtt{0}}(\pi(Z_{\mu}))=\operatorname{\mathtt{Res}}(\gamma(Z_{\mu}))$.
The left side is just $\gamma^{\mathtt{0}}(\mathscr{D}_{\mu})$. The right
side, when applied to $2\overline{\lambda^{\natural}}+\rho$ is just
$\gamma(Z_{\mu})(2\lambda^{\natural}_{\mathfrak{h}}+\rho_{\mathfrak{h}})$ by
Eq. (42) (c.f. Eq. (46)). Hence

(56)
$\gamma^{\mathtt{0}}(\mathscr{D}_{\mu})(2\overline{\lambda^{\natural}}+\rho)=\gamma(Z_{\mu})(2\lambda^{\natural}_{\mathfrak{h}}+\rho).$

By Theorem C, $Z_{\mu}$ acts by 0 on $I_{\lambda}$ for all
$\lambda\nsupseteq\mu$, $\lambda\in\mathscr{H}$, which exactly gives
$\gamma(Z_{\mu})(2\lambda^{\natural}_{\mathfrak{h}}+\rho)=0$. Thus

$\gamma^{\mathtt{0}}(\mathscr{D}_{\mu})(2\overline{\lambda^{\natural}}+\rho)=0$

for all $\lambda\nsupseteq\mu$, $\lambda\in\mathscr{H}$, proving the vanishing
properties.

Alternatively, by Proposition 5.8, we have the “relaxed” vanishing properties
$\gamma^{\mathtt{0}}(\mathscr{D}_{\mu})(2\overline{\lambda^{\natural}}+\rho)=0$
when$\lambda\neq\mu$, $|\lambda|\leq|\mu|$. By Proposition 6.2, this also
implies the full “extra” vanishing properties.

Therefore, $\gamma^{\mathtt{0}}(\mathscr{D}_{\mu})$ is proportional to
$J_{\mu}$ by its vanishing properties.

Step 2. We first pick a basis for $\mathfrak{p}^{+}$ extended from

$\\{\mathsf{i}E_{i,p+i},\mathsf{i}D_{j,q+j}\\},$

and a dual basis for $\mathfrak{p}^{-}\cong(\mathfrak{p}^{+})^{*}$ (identified
via the bilinear form $b$) extended from

$\\{-2\mathsf{i}E_{i,p+i},2\mathsf{i}D_{j,q+j}\\}.$

Then the identity map
$1_{\mathfrak{p}^{+}}\in\operatorname{End}_{\mathfrak{k}}(\mathfrak{p}^{+})$
corresponds to

(57)
$\sum_{i=1}^{p}\mathsf{i}E_{i,p+i}\otimes(-2\mathsf{i}E_{i,p+i})+\sum_{j=1}^{q}\mathsf{i}D_{j,p+j}\otimes(2\mathsf{i}D_{j,p+j})+\text{other
terms}$

in $\mathfrak{p}^{-}\otimes\mathfrak{p}^{+}$. On the other hand, we may extend
the basis $\\{x_{i},y_{j}\\}$ for $\mathfrak{a}$ (see Subsection 3.1) to a
basis for $\mathfrak{g}$ according to
$\mathfrak{g}=\mathfrak{n}^{-}\oplus\mathfrak{a}\oplus\mathfrak{k}$. This is
used to define ${\Gamma}$ and $\gamma^{\mathtt{0}}$. Then it is a direct
computation to see that the top homogeneous degree of

${\Gamma}(D_{(1)})=\gamma^{\mathtt{0}}(\mathscr{D}_{(1)})$

is precisely

$\frac{1}{2}\sum_{i=1}^{p}x_{i}^{2}-\frac{1}{2}\sum_{j=1}^{q}y_{j}^{2}.$

Note the $\rho$ shift does not change this top degree.

Now we let $\mu\in\mathscr{H}^{m}$. The identity map $1_{\mu}$ on $W_{\mu}$
corresponds to an element in $W_{\mu}^{*}\otimes W_{\mu}$. Thus the sum
$\sum_{\mu\in\mathscr{H}^{m}}1_{\mu}$ corresponds to an element in
$\mathfrak{S}^{m}(\mathfrak{p}^{-})\otimes\mathfrak{S}^{m}(\mathfrak{p}^{+})$
by the decomposition in Proposition 3.1. On
$\mathfrak{S}^{m}(\mathfrak{p})\subseteq T^{m}(\mathfrak{p})$, the bilinear
form is induced from $b$ on $\mathfrak{p}$ by normalizing $b^{\otimes m}$ by
$1/m!$. Then similarly, we see that the top homogeneous degree of
$\gamma^{\mathtt{0}}\left(\sum_{\mu\in\mathscr{H}^{m}}\mathscr{D}_{\mu}\right)$
is given by

(58)
$\frac{1}{m!}\left(\frac{1}{2}\sum_{i=1}^{p}x_{i}^{2}-\frac{1}{2}\sum_{j=1}^{q}y_{j}^{2}\right)^{m}=\frac{1}{2^{m}m!}\left(p_{2}^{(m,n)}(x_{i},y_{j})\right)^{m}.$

Note $d_{\lambda}$ (Eq. (12)) specializes to $(-1)^{|\lambda|}$. From Eq.
(11), Theorems 2.6 and 2.7, we see that

$I^{\textup{SV}}_{\mu^{\prime}}\left(x_{i},y_{j};-1,q-p+\frac{1}{2}\right)=(-1)^{|\mu|}\hat{I}_{\mu}\left(x_{i},y_{j};-1,\frac{1}{2}-(q-p)\right).$

and the top homogeneous degree of is
$(-1)^{|\mu|}SP_{\mu}(x_{i}^{2},y_{j}^{2};1)$. For $J_{\mu}$, this becomes

$(-1/2)^{|\mu|}SP_{\mu}(x_{i}^{2},y_{j}^{2};1).$

Thus by setting $\theta=1$ in Eq. (16),

(59)
$\frac{1}{2^{m}m!}\left(\sum_{i=1}^{p}x_{i}^{2}-\sum_{j=1}^{q}y_{j}^{2}\right)^{m}=\sum_{\mu\in\mathscr{H}^{m}}2^{-m}C_{\mu}^{-}(1;-1)(-2)^{|\mu|}\left((-1/2)^{|\mu|}SP_{\mu}(x_{i}^{2},y_{j}^{2};1)\right).$

Comparing the coefficients, we have

(60)
$\gamma^{\mathtt{0}}(\mathscr{D}_{\mu})=(-1)^{|\mu|}C_{\mu}^{-}(1;-1)J_{\mu}.$

Therefore the coefficient is precisely $k_{\mu}$ given by Eq. (55). ∎

Additionally, we can answer the interesting question of by which scalar
$D_{\mu}$ acts on the spherical vector $\omega\in I_{\mu}^{\mathfrak{k}}$,
without any direct consideration on the module $I_{\mu}$ itself!

######  Corollary 6.3.

The operator $D_{\mu}\in\mathfrak{U}^{\mathfrak{k}}$ acts on
$I_{\mu}^{\mathfrak{k}}$ by
$(-1)^{|\mu|}C_{\mu}^{-}(1;-1)^{2}C_{\mu}^{+}(2q-2p;-1)$.

###### Proof.

This scalar is the value of
${\Gamma}(D_{\mu})=\gamma^{\mathtt{0}}(\mathscr{D}_{\mu})$ at
$2\overline{\mu^{\natural}}+\rho$ which equals the value of
$k_{\mu}I^{\textup{SV}}_{\mu^{\prime}}$ at
$(z(\mu^{\prime}),w(\mu^{\prime}))$:

$\displaystyle\gamma^{\mathtt{0}}(\mathscr{D}_{\mu})(2\overline{\mu^{\natural}}+\rho)$
$\displaystyle=k_{\mu}C_{\mu^{\prime}}^{-}(1;-1)C_{\mu^{\prime}}^{+}(2q-2p;-1)$
$\displaystyle=(-1)^{|\mu|}C_{\mu}^{-}(1;-1)^{2}C_{\mu}^{+}(2q-2p;-1).$

∎

## References

  * [AHZ10] Alexander Alldridge, Joachim Hilgert, and Martin R. Zirnbauer, _Chevalley’s restriction theorem for reductive symmetric superpairs_ , J. Algebra 323 (2010), no. 4, 1159–1185. MR 2578599
  * [All12] A. Alldridge, _The Harish-Chandra isomorphism for reductive symmetric superpairs_ , Transform. Groups 17 (2012), no. 4, 889–919. MR 3000475
  * [AS15] Alexander Alldridge and Sebastian Schmittner, _Spherical representations of Lie supergroups_ , J. Funct. Anal. 268 (2015), no. 6, 1403–1453. MR 3306354
  * [ASS18] Alexander Alldridge, Siddhartha Sahi, and Hadi Salmasian, _Schur $Q$-functions and the Capelli eigenvalue problem for the Lie superalgebra $\mathfrak{q}(n)$_, Representation theory and harmonic analysis on symmetric spaces, Contemp. Math., vol. 714, Amer. Math. Soc., [Providence], RI, [2018] ©2018, pp. 1–21. MR 3847241
  * [CW01] Shun-Jen Cheng and Weiqiang Wang, _Howe duality for Lie superalgebras_ , Compositio Math. 128 (2001), no. 1, 55–94. MR 1847665
  * [CW12] by same author, _Dualities and representations of Lie superalgebras_ , Graduate Studies in Mathematics, vol. 144, American Mathematical Society, Providence, RI, 2012. MR 3012224
  * [FK90] J. Faraut and A. Korányi, _Function spaces and reproducing kernels on bounded symmetric domains_ , J. Funct. Anal. 88 (1990), no. 1, 64–89. MR 1033914
  * [Gor04] Maria Gorelik, _The Kac construction of the centre of $U(\mathfrak{g})$ for Lie superalgebras_, J. Nonlinear Math. Phys. 11 (2004), no. 3, 325–349. MR 2084313
  * [Hel00] Sigurdur Helgason, _Groups and geometric analysis_ , Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, Providence, RI, 2000, Integral geometry, invariant differential operators, and spherical functions, Corrected reprint of the 1984 original. MR 1790156
  * [Hum78] James E. Humphreys, _Introduction to Lie algebras and representation theory_ , Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978, Second printing, revised. MR 499562
  * [Kac84] Victor G. Kac, _Laplace operators of infinite-dimensional Lie algebras and theta functions_ , Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 2, 645–647. MR 735060
  * [Kos59] Bertram Kostant, _A formula for the multiplicity of a weight_ , Trans. Amer. Math. Soc. 93 (1959), 53–73. MR 109192
  * [KS93] Bertram Kostant and Siddhartha Sahi, _Jordan algebras and Capelli identities_ , Invent. Math. 112 (1993), no. 3, 657–664. MR 1218328
  * [Kum10] Shrawan Kumar, _Tensor product decomposition_ , Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1226–1261. MR 2827839
  * [LSS22a] Gail Letzter, Siddhartha Sahi, and Hadi Salmasian, _The Capelli eigenvalue problem for quantum groups_ , 2022, arXiv.
  * [LSS22b] by same author, _Weyl algebras for quantum homogeneous spaces_ , 2022, arXiv.
  * [LSS23] by same author, _Quantized Weyl algebras, the double centralizer property, and a new First Fundamental Theorem for $U_{q}(\mathfrak{gl}_{n})$_, 2023, arXiv.
  * [Mac95] I. G. Macdonald, _Symmetric functions and Hall polynomials_ , second ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, With contributions by A. Zelevinsky, Oxford Science Publications. MR 1354144
  * [Mus12] Ian M. Musson, _Lie superalgebras and enveloping algebras_ , Graduate Studies in Mathematics, vol. 131, American Mathematical Society, Providence, RI, 2012. MR 2906817
  * [Oko98] A. Okounkov, _${\rm BC}$ -type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials_, Transform. Groups 3 (1998), no. 2, 181–207. MR 1628453
  * [OO06] Andrei Okounkov and Grigori Olshanski, _Limits of $BC$-type orthogonal polynomials as the number of variables goes to infinity_, Jack, Hall-Littlewood and Macdonald polynomials, Contemp. Math., vol. 417, Amer. Math. Soc., Providence, RI, 2006, pp. 281–318. MR 2284134
  * [Sah94] Siddhartha Sahi, _The spectrum of certain invariant differential operators associated to a Hermitian symmetric space_ , Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 569–576. MR 1327549
  * [Sch70] Wilfried Schmid, _Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen_ , Invent. Math. 9 (1969/70), 61–80. MR 259164
  * [Ser99] Alexander Sergeev, _The invariant polynomials on simple Lie superalgebras_ , Represent. Theory 3 (1999), 250–280. MR 1714627
  * [She22] Alexander Sherman, _Iwasawa decomposition for Lie superalgebras_ , J. Lie Theory 32 (2022), no. 4, 973–996. MR 4512155
  * [Shi90] Goro Shimura, _Invariant differential operators on Hermitian symmetric spaces_ , Ann. of Math. (2) 132 (1990), no. 2, 237–272. MR 1070598
  * [SS16] Siddhartha Sahi and Hadi Salmasian, _The Capelli problem for $\mathfrak{gl}(m|n)$ and the spectrum of invariant differential operators_, Adv. Math. 303 (2016), 1–38. MR 3552519
  * [SSS20] Siddhartha Sahi, Hadi Salmasian, and Vera Serganova, _The Capelli eigenvalue problem for Lie superalgebras_ , Math. Z. 294 (2020), no. 1-2, 359–395. MR 4054814
  * [Sta89] Richard P. Stanley, _Some combinatorial properties of Jack symmetric functions_ , Adv. Math. 77 (1989), no. 1, 76–115. MR 1014073
  * [Ste85] John R. Stembridge, _A characterization of supersymmetric polynomials_ , J. Algebra 95 (1985), no. 2, 439–444. MR 801279
  * [SV05] A. N. Sergeev and A. P. Veselov, _Generalised discriminants, deformed Calogero-Moser-Sutherland operators and super-Jack polynomials_ , Adv. Math. 192 (2005), no. 2, 341–375. MR 2128703
  * [SV09] Alexander N. Sergeev and Alexander P. Veselov, _$BC_{\infty}$ Calogero-Moser operator and super Jacobi polynomials_, Adv. Math. 222 (2009), no. 5, 1687–1726. MR 2555909
  * [SV11] by same author, _Grothendieck rings of basic classical Lie superalgebras_ , Ann. of Math. (2) 173 (2011), no. 2, 663–703. MR 2776360
  * [SZ19] Siddhartha Sahi and Genkai Zhang, _Positivity of Shimura operators_ , Math. Res. Lett. 26 (2019), no. 2, 587–626. MR 3999556
  * [Yan92] Zhi Min Yan, _A class of generalized hypergeometric functions in several variables_ , Canad. J. Math. 44 (1992), no. 6, 1317–1338. MR 1192421
  * [Zhu22] Songhao Zhu, _Shimura operators for certain Hermitian symmetric superpairs_ , arXiv, submitted, 2022.