Supersymmetric Shimura operators and interpolation polynomials

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.

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].

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 $\mathfrak{a}$.

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].

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)

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

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].

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.

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.

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

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

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)

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

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)

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