The Capelli eigenvalue problem for Lie superalgebras

Abstract

For a finite dimensional unital complex simple Jordan superalgebra J, the Tits–Kantor–Koecher construction yields a 3-graded Lie superalgebra $\mathfrak {g}^\flat \cong \mathfrak {g}^\flat (-1)\oplus \mathfrak {g}^\flat (0)\oplus \mathfrak {g}^\flat (1)$, such that $\mathfrak {g}^\flat (-1)\cong J$. Set $V:=\mathfrak {g}^\flat (-1)^*$ and $\mathfrak {g}:=\mathfrak {g}^\flat (0)$. In most cases, the space $\mathscr {P}(V)$ of superpolynomials on V is a completely reducible and multiplicity-free representation of $\mathfrak {g}$, and there exists a direct sum decomposition $\mathscr {P}(V):=\bigoplus _{\lambda \in \Omega }V_\lambda$, where $\left( V_\lambda \right) _{\lambda \in \Omega }$ is a family of irreducible $\mathfrak {g}$-modules parametrized by a set of partitions $\Omega$. In these cases, one can define a natural basis $\left( D_\lambda \right) _{\lambda \in \Omega }$ of “Capelli operators” for the algebra $\mathscr {PD}(V)^{\mathfrak {g}}$ of $\mathfrak {g}$-invariant superpolynomial differential operators on V. In this paper we complete the solution to the Capelli eigenvalue problem, which asks for the determination of the scalar $c_\mu (\lambda )$ by which $D_\mu$ acts on $V_\lambda$. We associate a restricted root system $\varSigma$ to the symmetric pair $(\mathfrak {g},\mathfrak {k})$ that corresponds to J, which is either a deformed root system of type $\mathsf {A}(m,n)$ or a root system of type $\mathsf {Q}(n)$. We prove a necessary and sufficient condition on the structure of $\varSigma$ for $\mathscr {P}(V)$ to be completely reducible and multiplicity-free. When $\varSigma$ satisfies the latter condition we obtain an explicit formula for the eigenvalue $c_\mu (\lambda )$, in terms of Sergeev–Veselov’s shifted super Jack polynomials when $\varSigma$ is of type $\mathsf {A}(m,n)$, and Okounkov-Ivanov’s factorial Schur Q-polynomials when $\varSigma$ is of type $\mathsf {Q}(n)$. Along the way, we prove that the natural map from the centre of the enveloping algebra of $\mathfrak {g}$ into $\mathscr {PD}(V)^{\mathfrak {g}}$ is surjective in all cases except when $J\cong F$, where $F$ is the 10-dimensional exceptional Jordan superalgebra.

Appendices

Appendix A. The TKK construction

Recall that a vector superspace $J:=J_{\overline{\mathbf {0}}}\oplus J_{\overline{\mathbf {1}}}$ is called a Jordan superalgebra if it is equipped with a supercommutative bilinear product $J\times J\rightarrow J$ which satisfies the Jordan identity

$$\begin{aligned}&(-1)^{|x||z|}[L_x,L_{yz}]+(-1)^{|y||x|}[L_y,L_{zx}]+(-1)^{|z||y|}[L_z,L_{xy}]=0 \ \text { for homogeneous }\\&\quad x,y,z\in J, \end{aligned}$$

where we define $L_a:J\rightarrow J$ for $a\in J$ to be the left multiplication map $x\mapsto ax$, and denote the parity of a homogeneous element $a\in J$ by |a|.

Following [3], by a short grading of a Lie superalgebra $\mathfrak {l}$ we mean a ${\mathbb {Z}}$-grading of $\mathfrak {l}$ of the form $\mathfrak {l}:=\bigoplus _{t\in {\mathbb {Z}}}\mathfrak {l}(t)$, such that $\mathfrak {l}(t)=\{0\}$ for $t\not \in \{0,\pm 1\}$. Using the Kantor functor, in [16] Kac associates to J a simple Lie superalgebra $\mathfrak {g}_J^{}$ (the TKK Lie superalgebra) with a short grading

$$\begin{aligned} \mathfrak {g}_J^{}:= \mathfrak {g}_J^{}(-1)\oplus \mathfrak {g}_J^{}(0)\oplus \mathfrak {g}_J^{}(1). \end{aligned}$$

We recall the definition of $\mathfrak {g}_J^{}$. Set $\mathfrak {g}_J^{}(-1):=J$, $\mathfrak {g}_J^{}(0):={\mathrm{Span}}_{\mathbb {C}}\{L_a,[L_a,L_b]\,:\,a,b\in J\}\subseteq \mathrm {End}_{\mathbb {C}}(J)$, and $\mathfrak {g}_J^{}(1):={\mathrm{Span}}_{\mathbb {C}}\{ P,[L_a,P]\,:\,a\in J\}\subseteq \mathrm {Hom}_{\mathbb {C}}(\mathcal {S}^2(J),J),$ where $P:\mathcal {S}^2(J)\rightarrow J$ is the map $P(x,y):=xy$, and $[L_a,P](x,y):=a(xy)-(ax)y-(-1)^{|x||y|}(ay)x$. The Lie superbracket of $\mathfrak {g}_J^{}$ is defined by the following relations.

• (i) $[A,a]:=A(a)$ for $A\in \mathfrak {g}_J^{}(0)$ and $a\in \mathfrak {g}_J^{}(-1)$.

• (ii) $[A,a](x):=A(a,x)$ for $A\in \mathfrak {g}_J^{}(1)$, $a\in \mathfrak {g}_J^{}(-1)$, and $x\in J$.

• (iii) $[A,B](x,y):=A(B(x,y))-(-1)^{|A||B|}B(A(x),y)-(-1)^{|A||B|+|x||y|}B(A(y),x)$ for $A\in \mathfrak {g}_J^{}(0)$, $B\in \mathfrak {g}_J^{}(1)$, and $x,y\in J$.

For the classification of finite dimensional complex simple Jordan superalgebras and their corresponding TKK Lie superalgebras, see the articles by Kac [16] and Cantarini and Kac [3].

If J has a unit $1_J\in J$, then the elements $e:=1_J$, $f:=-2P$, and $h:=2L_{1_J}$ of $\mathfrak {g}_J^{}$ satisfy (1). It follows that $\mathfrak {s}:={\mathrm{Span}}_{\mathbb {C}}\{e,f,h\}$ is a subalgebra of $\mathfrak {g}_J^{}$ isomorphic to $\mathfrak {sl}_2({\mathbb {C}})$. Indeed $\mathfrak {s}$ is a short subalgebra of $\mathfrak {g}_J^{}$ (see [3]). We recall the definition of a short subalgebra.

Definition A.1

Let $\mathfrak {l}$ be a complex Lie superalgebra. A short subalgebra of $\mathfrak {l}$ is a Lie subalgebra $\mathfrak {a}\subseteq \mathfrak {l}_{\overline{\mathbf {0}}}$ that is isomorphic to $\mathfrak {sl}_2({\mathbb {C}})$, with a basis e, f, h that satisfies the relations (1), such that the eigenspace decomposition of $\mathrm {ad}\left( -\frac{1}{2}h\right)$ defines a short grading of $\mathfrak {l}$.

Remark A.2

Let $\mathfrak {l}$ be a complex Lie superalgebra and let $\mathfrak {a}\subseteq \mathfrak {l}_{\overline{\mathbf {0}}}$ be a short subalgebra of $\mathfrak {l}$.

• (a) Assume that $\mathfrak {l}$ is a subalgebra of another Lie superalgebra $\tilde{\mathfrak {l}}$ such that $\dim \tilde{\mathfrak {l}}=\dim \mathfrak {l}+1$. Since every finite dimensional $\mathfrak {sl}_2({\mathbb {C}})$-module is completely reducible, it follows that $\tilde{\mathfrak {l}}\cong \mathfrak {l}\oplus {\mathbb {C}}$ as $\mathfrak {a}$-modules, so that $\mathfrak {a}$ is a short subalgebra of $\tilde{\mathfrak {l}}$ as well.

• (b) Every central extension $0\rightarrow {\mathbb {C}}\rightarrow \hat{\mathfrak {l}}\rightarrow \mathfrak {l}\rightarrow 0$ splits on $\mathfrak {a}$. An argument similar to part (a) implies that the image of $\mathfrak {a}$ under the splitting section is a short subalgebra of $\hat{\mathfrak {l}}$.

When J is isomorphic to $gl (m,n)_+$, $p (n)_+$, or $q (n)_+$, it will be more convenient for us to replace $\mathfrak {g}_J^{}$ by a non-simple Lie superalgebra which has a more natural matrix realization (see also Remark A.3). To this end, we define the Lie superalgebra $\mathfrak {g}^\flat$ by

$$\begin{aligned} \mathfrak {g}^\flat := {\left\{ \begin{array}{ll} \mathfrak {gl}(2m|2n)&{}\text { if } J\cong gl (m,n)_+,\\ \mathfrak {p}(2n)&{}\text { if }J\cong p (n)_+,\\ \mathfrak {q}(2n)&{}\text { if }J\cong q (n)_+,\\ \mathfrak {g}_J^{}&{}\text { otherwise.} \end{array}\right. } \end{aligned}$$

(54)

For a precise description of $\mathfrak {p}(2n)$ and $\mathfrak {q}(2n)$ see Appendix B. From Remark A.2 it follows that the short subalgebra $\mathfrak {s}$ of $\mathfrak {g}_J^{}$ corresponds to a unique short subalgebra of $\mathfrak {g}^\flat$. We use the same symbols $\mathfrak {s}$, e, f, and h for denoting the short subalgebra of $\mathfrak {g}^\flat$ and its corresponding basis.

By restriction of the adjoint representation, $\mathfrak {g}^\flat$ is equipped with an $\mathfrak {s}$-module structure. This $\mathfrak {s}$-module is a direct sum of trivial and adjoint representations of $\mathfrak {s}$, hence it integrates to a representation of the adjoint group $\mathrm {PSL}_2({\mathbb {C}})$. Furthermore,

$$\begin{aligned} w:=\exp (\mathrm {ad}({f})) \exp (-\mathrm {ad}({e})) \exp (\mathrm {ad}({f})) \end{aligned}$$

(55)

represents the nontrivial element of the Weyl group of $\mathrm {PSL}_2({\mathbb {C}})$.

Set $\mathfrak {g}^\flat (t):=\{x\in \mathfrak {g}^\flat \, :\, [h,x]=-2tx\}$ for $t\in \{0,\pm 1\}$. The Lie superalgebra $\mathfrak {g}^\flat (0)$ naturally acts on $\mathfrak {g}^\flat (-1)\cong J$. Set

$$\begin{aligned} \mathfrak {g}:=\mathfrak {g}^\flat (0) \text { and } V:=\mathfrak {g}^\flat (-1)^*:=\mathrm {Hom}_{\mathbb {C}}(V,{\mathbb {C}}^{1|0}). \end{aligned}$$

Thus the $\mathfrak {g}$-module V is the dual of the $\mathfrak {g}$-module J.

Remark A.3

The reason for replacing $\mathfrak {g}_J^{}$ by $\mathfrak {g}^\flat$ is to obtain a convenient way of associating partitions to the irreducible $\mathfrak {g}$-modules that occur in $\mathscr {P}(V)$. For example, assume that $J\cong gl (m,n)_+$, where $gl (m,n)_+$ denotes the Jordan superalgebra of $(m+n)\times (m+n)$ matrices in (m, n)-block form. Then $\mathfrak {g}_J^{}\cong \mathfrak {sl}(2m|2n)$ if $m\ne n$, and $\mathfrak {g}_J^{}\cong \mathfrak {psl}(2m|2n)$ if $m=n$. In both cases, $\mathfrak {g}_J^{}(0)$ is closely related to $\mathfrak {gl}(m|n)\oplus \mathfrak {gl}(m|n)$, but it is not isomorphic to it. However, $\mathfrak {g}:= \mathfrak {g}^\flat (0)\cong \mathfrak {gl}(m|n)\oplus \mathfrak {gl}(m|n)$, and the irreducible summands of $\mathscr {P}(V)\cong \mathscr {P}(({\mathbb {C}}^{m|n})^*\otimes {\mathbb {C}}^{m|n})$ are naturally parametrized by (m, n)-hook partitions.

Appendix B. Classical Lie superalgebras

In this Appendix we give explicit realizations of classical Lie superalgebras $\mathfrak {gl}(m|n)$, $\mathfrak {gosp}(m|2n)$, $\mathfrak {p}(n)$, and $\mathfrak {q}(n)$. We describe root systems of $\mathfrak {gl}(m|n)$, $\mathfrak {gosp}(m|2n)$, and $\mathfrak {q}(n)$, and choose Borel subalgebras in these Lie superalgebras.

1.1 B.1. The Lie superalgebra $\mathfrak {gl}(m|n)$

Let $m,n\ge 1$ be integers. We use the usual realization of $\mathfrak {gl}(m|n)$ as $(m+n)\times (m+n)$ matrices in (m, n)-block form

$$\begin{aligned} \begin{bmatrix} A&B\\ C&D \end{bmatrix} \end{aligned}$$

(56)

where A is $m\times m$ and D is $n\times n$. The diagonal Cartan subalgebra of $\mathfrak {gl}(m|n)$ is

$$\begin{aligned} \mathfrak {h}_{m|n}:=\left\{ \mathrm {diag}({\mathbf {s}},{\mathbf {t}})\,:\,\mathbf {s}:=(s_1,\ldots , s_m)\in {\mathbb {C}}^m\text { and }\mathbf {t}:=(t_1,\ldots ,t_n)\in {\mathbb {C}}^n\right\} . \end{aligned}$$

(57)

The standard characters $\varepsilon _i,\delta _j:\mathfrak {h}_{m|n}\rightarrow {\mathbb {C}}$ are defined by

$$\begin{aligned} \varepsilon _i(\mathrm {diag}({\mathbf {s}},{\mathbf {t}})):=s_i\text { for }1\le i\le m\ \text { and }\ \delta _j(\mathrm {diag}({\mathbf {s}},{\mathbf {t}})):=t_j\text { for }1\le j\le n. \end{aligned}$$

We define $\mathfrak {b}^{\mathrm {st}}_{m|n}$ (respectively, $\mathfrak {b}^{\mathrm {op}}_{m|n}$) to be the Borel subalgebras of $\mathfrak {gl}(m|n)$ corresponding to the fundamental systems ${\Pi }^{\mathrm {st}}$ (respectively, ${\Pi }^{\mathrm {op}}$), where

$$\begin{aligned} {\Pi }^{\mathrm {st}}:=\left\{ \varepsilon _i-\varepsilon _{i+1}\right\} _{i=1}^{m-1}\cup \left\{ \varepsilon _m-\delta _1\right\} \cup \left\{ \delta _j-\delta _{j+1}\right\} _{j=1}^{n-1} \text { and } {\Pi }^{\mathrm {op}}:=-{\Pi }^{\mathrm {st}} \end{aligned}$$

For every partition $\lambda \in \mathcal {H}(m,n)$, we set

$$\begin{aligned} \lambda ^{\mathrm {st}}_{m|n}:= \sum _{i=1}^m\lambda _i\varepsilon _i+ \sum _{j=1}^n\langle \lambda _j'-m\rangle \delta _j. \end{aligned}$$

(58)

In the spacial case $m=n$, we define $\mathfrak {b}_{n|n}^{\mathrm {mx}}$ to be the Borel subalgebra corresponding to the fundamental system

$$\begin{aligned} {\Pi }^{\mathrm {mx}} :=\left\{ \delta _i-\varepsilon _i \right\} _{i=1}^n\cup \left\{ \varepsilon _{j}-\delta _{j-1} \right\} _{j=2}^{n}. \end{aligned}$$

1.2 B.2. The Lie superalgebra $\mathfrak {gosp}(m|2n)$

Let $m,n\ge 1$ be integers. We begin with an explicit realization of $\mathfrak {osp}(m|2n)$. Set $r:=\lfloor \frac{m}{2}\rfloor$. Let $J^+$ be the $m\times m$ matrix defined by

$$\begin{aligned} J^+:=\begin{bmatrix} 1&0_{1\times r}&0_{1\times r}\\ 0_{r\times 1}&0_{r\times r}&I_{r\times r}\\ 0_{r\times 1}&I_{r\times r}&0_{r\times r} \end{bmatrix} \text { if } m=2r+1, \text { and } J^+:=\begin{bmatrix} 0_{r\times r}&I_{r\times r}\\ I_{r\times 1}&0_{r\times r} \end{bmatrix} \text { if } m=2r, \end{aligned}$$

Also, let $J^-$ be the $2n\times 2n$ matrix defined by

$$\begin{aligned} J^-:=\begin{bmatrix} 0_{n\times n}&I_{n\times n}\\ -I_{n\times n}&0_{n\times n} \end{bmatrix}. \end{aligned}$$

Let $\{\mathsf {e}_i\}_{i=1}^m\cup \{\mathsf {e}'_j\}_{j=1}^{2n}$ be the standard homogeneous basis of ${\mathbb {C}}^{m|2n}$, and let $\mathsf {B}:{\mathbb {C}}^{m|2n}\times {\mathbb {C}}^{m|2n}\rightarrow {\mathbb {C}}$ be the even supersymmetric bilinear form defined by

$$\begin{aligned} \mathsf {B}(\mathsf {e}_i,\mathsf {e}_j)=J^+_{i,j},\ \mathsf {B}(\mathsf {e}_i',\mathsf {e}_j')=J^-_{i,j}, \text { and } \mathsf {B}(\mathsf {e}_i,\mathsf {e}_j')=0. \end{aligned}$$

We realize the Lie superalgebra $\mathfrak {osp}(m|2n)$ as the subalgebra of $\mathfrak {gl}(m|2n)$ that leaves the bilinear form ${\mathsf {B}}:{\mathbb {C}}^{m|2n}\times {\mathbb {C}}^{m|2n}\rightarrow {\mathbb {C}}$ invariant. For ${\mathbf {s}}\in {\mathbb {C}}^r$ and ${\mathbf {t}}\in {\mathbb {C}}^n$, set

$$\begin{aligned} {\mathsf {d}}({\mathbf {s}},{\mathbf {t}}):= {\left\{ \begin{array}{ll} \mathrm {diag}({\mathbf {s}},-{\mathbf {s}},{\mathbf {t}},-{\mathbf {t}})&{}\text { if }m=2r,\\ \mathrm {diag}(0,{\mathbf {s}},-{\mathbf {s}},{\mathbf {t}},-{\mathbf {t}})&{}\text { if }m=2r+1. \end{array}\right. } \end{aligned}$$

Recall from Appendix B.1 that we denote the standard Cartan subalgebra of $\mathfrak {gl}(m|2n)$ by $\mathfrak {h}_{m|2n}$. Then $\overline{\mathfrak {h}}_{m|2n}:=\mathfrak {h}_{m|2n}\cap \mathfrak {osp}(m|2n)$ is a Cartan subalgebra of $\mathfrak {osp}(m|2n)$. We have

$$\begin{aligned} \overline{\mathfrak {h}}_{m|2n}=\left\{ {\mathsf {d}}({\mathbf {s}},{\mathbf {t}})\,:\,{\mathbf {s}}\in {\mathbb {C}}^r\text { and }{\mathbf {t}}\in {\mathbb {C}}^n\right\} , \end{aligned}$$

and the standard characters of $\overline{\mathfrak {h}}_{m|2n}$ are given by

$$\begin{aligned} \varepsilon _i({\mathsf {d}}({\mathbf {s}},{\mathbf {t}})):=s_i \text { for }1\le i\le r \text { and } \delta _j({\mathsf {d}}({\mathbf {s}},{\mathbf {t}})):=t_j \text { for }1\le j\le n. \end{aligned}$$

Let $\breve{\mathfrak {b}}_{m|2n}\subseteq \mathfrak {osp}(m|2n)$ be the Borel subalgebra corresponding to the fundamental system ${\Pi }$, where

$$\begin{aligned} {\Pi } := {\left\{ \begin{array}{ll} \left\{ \varepsilon _i-\varepsilon _{i+1} \right\} _{i=1}^{r-1} \cup \left\{ \varepsilon _r-\delta _1\right\} \cup \left\{ \delta _j-\delta _{j+1} \right\} _{j=1}^{n-1} \cup \left\{ \delta _n \right\} &{}\text { if }m=2r+1,\\ \left\{ \varepsilon _i-\varepsilon _{i+1} \right\} _{i=1}^{r-1} \cup \left\{ \varepsilon _r-\delta _1\right\} \cup \left\{ \delta _j-\delta _{j+1} \right\} _{j=1}^{n-1} \cup \left\{ 2\delta _n \right\} &{}\text { if }m=2r. \end{array}\right. } \end{aligned}$$

Finally, we set $\mathfrak {gosp}(m|2n):=\mathfrak {osp}(m|2n)\oplus {\mathbb {C}}I\subseteq \mathfrak {gl}(m|2n)$, where $I:=I_{(m+2n)\times (m+2n)}$. We also set

$$\begin{aligned} \mathfrak {b}_{m|2n}:=\breve{\mathfrak {b}}_{m|2n}\oplus {\mathbb {C}}I. \end{aligned}$$

(59)

We extend the standard characters $\varepsilon _i,\delta _j$ of $\overline{\mathfrak {h}}_{m|2n}$ to the subalgebra $\tilde{\mathfrak {h}}_{m|2n}:=\mathfrak {h}_{m|2n}\cap \mathfrak {gosp}(m|2n)$ of diagonal matrices in $\mathfrak {gosp}(m|2n)$, by setting $\varepsilon _i(I)=\delta _j(I)=0$. Let $\zeta :\tilde{\mathfrak {h}}_{m|2n}\rightarrow {\mathbb {C}}$ be the linear functional defined uniquely by

$$\begin{aligned} \zeta \big |_{\overline{\mathfrak {h}}_{m|2n}}=0\text { and }\zeta (I)=1. \end{aligned}$$

The set $\{\varepsilon _i\}_{i=1}^r\cup \{\delta _j\}_{j=1}^n\cup \{\zeta \}$ is a basis for the dual of $\tilde{\mathfrak {h}}_{m|2n}$. We remark that when $\mathfrak {g}$ is of type $\mathfrak {gosp}$ (i.e., in Cases III and V), we have $\zeta (h)=2$ where $h\in \mathfrak {g}^\flat$ is defined as in (1).

In Case V, where $\mathfrak {g}=\mathfrak {gosp}(2|4)$, we need to consider an exceptional Borel subalgebra

$$\begin{aligned} \mathfrak {b}_{2|4}^\mathrm {ex}:= \hat{\mathfrak {b}}^{}_{2|4}\oplus {\mathbb {C}}I, \end{aligned}$$

(60)

where $\hat{\mathfrak {b}}^{}_{2|4}$ is the Borel subalgebra of $\mathfrak {osp}(2|4)$ corresponding to the fundamental system

$$\begin{aligned} {\Pi }^{\mathrm {ex}}:= \{-\varepsilon _1-\delta _1,\delta _1-\delta _2,2\delta _2\}. \end{aligned}$$

1.3 B.3. The anisotropic embedding of $\mathfrak {osp}(m|2n)$ in $\mathfrak {gl}(m|2n)$

We will need another realization of $\mathfrak {osp}(m|2n)$ inside $\mathfrak {gl}(m|2n)$ which will be used in the description of the spherical subalgebra $\mathfrak {k}$. Set $r:=\lfloor \frac{m}{2}\rfloor$, and let $\tilde{J}^-$ be the $2n\times 2n$ matrix defined by

$$\begin{aligned} \tilde{J}^-:= \mathrm {diag} (\underbrace{\tilde{J},\ldots ,\tilde{J}}_{n\text { times}}) \text { where } \tilde{J}:=\begin{bmatrix} 0&1\\ -1&0\end{bmatrix} . \end{aligned}$$

Let $\tilde{\mathsf {B}}:{\mathbb {C}}^{m|2n}\times {\mathbb {C}}^{m|2n}\rightarrow {\mathbb {C}}$ be the even supersymmetric bilinear form which is given in the standard basis $\{\mathsf {e}_i\}_{i=1}^m\cup \{\mathsf {e}_j\}_{j=1}^{2n}$ of ${\mathbb {C}}^{m|2n}$ by

$$\begin{aligned} \tilde{\mathsf {B}}(\mathsf {e}_i,\mathsf {e}_j):=\delta _{i,j},\ \tilde{\mathsf {B}}(\mathsf {e}'_i,\mathsf {e}'_j):=\tilde{J}^-_{i,j}, \text { and } \tilde{\mathsf {B}}(\mathsf {e}_i,\mathsf {e}_j'):=0. \end{aligned}$$

Thus, the matrix of $\tilde{\mathsf {B}}(\cdot ,\cdot )$ in the standard basis of ${\mathbb {C}}^{m|2n}$ is

$$\begin{aligned} \begin{bmatrix}I_{m\times m}&0_{m\times 2n}\\0_{2n\times m}&\tilde{J}^-\end{bmatrix}. \end{aligned}$$

The subalgebra of $\mathfrak {gl}(m|2n)$ that leaves the bilinear form $\tilde{\mathsf {B}}$ invariant is isomorphic to $\mathfrak {osp}(m|2n)$.

1.4 B.4. The exceptional embedding of $\mathfrak {osp}(1|2)\oplus \mathfrak {osp}(1|2)$ in $\mathfrak {gosp}(2|4)$

We consider the realization of $\mathfrak {osp}(2|4)$ given in Appendix  B.2. Set

$$\begin{aligned} g:=\begin{bmatrix} 0&\sqrt{-1}&0&0&0&0\\ -\sqrt{-1}&0&0&0&0&0\\ 0&0&0&0&0&-\sqrt{-1}\\ 0&0&0&0&\sqrt{-1}&0\\ 0&0&0&-\sqrt{-1}&0&0\\ 0&0&\sqrt{-1}&0&0&0 \end{bmatrix}. \end{aligned}$$

We set $\mathfrak {k}^\mathrm {ex}$ to be the subalgebra of fixed points of the map $\mathfrak {osp}(2|4)\rightarrow \mathfrak {osp}(2|4)$ given by $x\mapsto \mathrm {Ad}_g(x)$. One can verify that $\mathfrak {k}^\mathrm {ex}\cong \mathfrak {osp}(1|2)\oplus \mathfrak {osp}(1|2)$. We will consider $\mathfrak {k}^\mathrm {ex}$ as a subalgebra of $\mathfrak {gosp}(2|4)$.

1.5 B.5. The Lie superalgebra $\mathfrak {p}(n)$

Let $n\ge 1$ be an integer, and let $\check{\mathsf {B}}:{\mathbb {C}}^{n|n}\times {\mathbb {C}}^{n|n}\rightarrow {\mathbb {C}}$ be the odd supersymmetric bilinear form defined by

$$\begin{aligned} \check{\mathsf {B}}(\mathsf {e}_i,\mathsf {e}_j'):=\delta _{i,j},\ \check{\mathsf {B}}(\mathsf {e}_i,\mathsf {e}_j):=0, \text { and } \check{\mathsf {B}}(\mathsf {e}'_i,\mathsf {e}'_j):=0, \end{aligned}$$

where $\{\mathsf {e}_i\}_{i=1}^n \cup \{\mathsf {e}_i'\}_{i=1}^n$ is the standard homogeneous basis of ${\mathbb {C}}^{n|n}$. The Lie superalgebra $\mathfrak {p}(n)$ is the subalgebra of $\mathfrak {gl}(n|n)$ that leaves $\check{\mathsf {B}}(\cdot ,\cdot )$ invariant. It consists of matrices in (n, n)-block form

$$\begin{aligned} \begin{bmatrix} A&B\\ C&-A^T \end{bmatrix},\text { where }B=B^T\text { and }C=-C^T. \end{aligned}$$

In this paper we will not need a description of the root system and highest weight modules of $\mathfrak {p}(n)$.

1.6 B.6. The Lie superalgebra $\mathfrak {q}(n)$

Let $n\ge 1$ be an integer. The Lie superalgebra $\mathfrak {q}(n)$ is the subalgebra of $\mathfrak {gl}(n|n)$ that consists of matrices in (n, n)-block form

$$\begin{aligned} \begin{bmatrix} A&B\\ B&A \end{bmatrix}. \end{aligned}$$

Let $\mathfrak {h}$ be the subalgebra of matrices of the latter form where A and B are diagonal. Then $\mathfrak {h}$ is a Cartan subalgebra of $\mathfrak {q}(n)$. The standard characters $\left\{ \varepsilon _i\right\} _{i=1}^n$ of $\mathfrak {h}_{\overline{\mathbf {0}}}$ are the restrictions of the corresponding standard characters of $\mathfrak {gl}(n|n)$. Let $\mathfrak {b}^{\mathrm {st}}_n$ (respectively, $\mathfrak {b}^{\mathrm {op}}_n$) be the Borel subalgebra of $\mathfrak {q}(n)$ associated to the fundamental system ${\Pi }^{\mathrm {st}}:= \{\varepsilon _i-\varepsilon _{i+1}\}_{i=1}^{n-1}$ (respectively, ${\Pi }^{\mathrm {op}}:= \{\varepsilon _{i+1}-\varepsilon _{i}\}_{i=1}^{n-1}$). For every partition $\lambda \in \mathcal {DP}(n)$, we set $\lambda ^{\mathrm {st}}_n:=\sum _{i=1}^n\lambda _i\varepsilon _i$.

Appendix C. Facts from supergeometry

All of the supermanifolds that are considered in this appendix are complex analytic. We denote the underlying complex manifold of a supermanifold $\mathcal {X}$ by $|\mathcal {X}|$, and the sheaf of superfunctions on $\mathcal {X}$ by $\mathcal {O}_{{\mathcal {X}}}$. Morphisms of supermanfolds are expressed as $(\mathsf {f},\mathsf {f}^\#):{\mathcal {X}}\rightarrow {\mathcal {Y}}$, where $\mathsf {f}:|{\mathcal {X}}|\rightarrow |{\mathcal {Y}}|$ is the complex analytic map between the underlying spaces and $\mathsf {f}^\#:\mathcal {O}_{{\mathcal {Y}}}\rightarrow \mathsf {f}_*\mathcal {O}_{{\mathcal {X}}}$ is the associated morphism of sheaves of superalgebras.

Let ${\mathcal {L}}$ be a connected Lie supergroup and let ${\mathcal {M}}$ be a Lie subsupergroup of ${\mathcal {L}}$. Set $\mathfrak {l}:=\mathrm {Lie}({\mathcal {L}})$ and $\mathfrak {m}:=\mathrm {Lie}({\mathcal {M}})$. The right action of ${\mathcal {L}}$ on ${\mathcal {L}}$ induces a canonical isomorphism of superalgebras from $\mathbf {U}(\mathfrak {l})$ onto the algebra of left invariant holomorphic differential operators on ${\mathcal {L}}$. Under this isomorphism elements of $\mathbf {U}(\mathfrak {l})^{\mathcal {M}}$, the subalgebra of ${\mathcal {M}}$-invariants in $\mathbf {U}(\mathfrak {l})$, are mapped to holomorphic differential operators which are left ${\mathcal {L}}$-invariant and right ${\mathcal {M}}$-invariant. The latter differential operators induce ${\mathcal {L}}$-invariant differential operators on the homogeneous space ${\mathcal {L}}/{\mathcal {M}}$. Consequently, we obtain a homomorphism of superalgebras

$$\begin{aligned} \Psi _{{\mathcal {L}},{\mathcal {M}}}:\mathbf {U}(\mathfrak {l})^{{\mathcal {M}}}\rightarrow \mathscr {D}({\mathcal {L}}/{\mathcal {M}}), \end{aligned}$$

(61)

where $\mathscr {D}({\mathcal {L}}/{\mathcal {M}})$ denotes the algebra of ${\mathcal {L}}$-invariant differential operators on ${\mathcal {L}}/{\mathcal {M}}$. By a superization of the argument of [20, Prop. 9.1], we obtain the following statement.

Proposition C.1

Let ${\mathscr {D}}^{(d)}({\mathcal {L}}/{\mathcal {M}})$ denote the subspace of elements of ${\mathscr {D}}({\mathcal {L}}/{\mathcal {M}})$ of order at most d. Assume that there exists an ${\mathcal {M}}$-invariant complement of $\mathfrak {m}$ in $\mathfrak {l}$. Then

$$\begin{aligned} \Psi _{{\mathcal {L}},{\mathcal {M}}}\left( \mathbf {U}^{(d)}(\mathfrak {l})^{\mathcal {M}}\right) = {\mathscr {D}}^{(d)}({\mathcal {L}}/{\mathcal {M}}) \text { for every } d\ge 0. \end{aligned}$$

In the rest of this appendix we will assume that J is a Jordan superalgebra of type $\mathsf {A}$. Let $\mathfrak {g}$, $\mathfrak {k}$, and V be as in Sect. 1.

Lemma C.2

There is a vector $v_{\mathfrak {k}}\in V_{\overline{\mathbf {0}}}$ such that $\mathfrak {k}=\mathrm {stab}_{\mathfrak {g}}(v_{\mathfrak {k}})$.

Proof

In all of the cases where J is of type $\mathsf {A}$ we have $V\cong V^*\cong J$ as $\mathfrak {k}$-modules, hence we can set $v_{\mathfrak {k}}$ equal to the element of $V_{\overline{\mathbf {0}}}$ corresponding to $1_J\in J$. $\square$

Lemma C.3

The map $\mathfrak {g}\rightarrow V$, $x\mapsto x\cdot v_{\mathfrak {k}}$ is surjective.

Proof

The kernel of the linear map $x\mapsto x\cdot v_{\mathfrak {k}}$ is $\mathfrak {k}$. The statement now follows in all of the cases by verifying that the graded dimension of the image of this map and of V are the same. $\square$

Let $\tilde{\mathfrak {b}}:=\tilde{\mathfrak {h}}\oplus \tilde{\mathfrak {n}}$ be a Borel subalgebra of $\mathfrak {g}$ such that $\mathfrak {g}=\tilde{\mathfrak {b}}+\mathfrak {k}$. Let ${\mathcal {G}}$ be a complex Lie supergroup such that $\mathrm {Lie}({\mathcal {G}})=\mathfrak {g}$, and let ${\mathcal {V}}$ be the complex affine superspace corresponding to V. We assume that $|{\mathcal {G}}|$ is a connected Lie group, and that the action of $\mathfrak {g}$ on V can be globalized to an action of ${\mathcal {G}}$ on ${\mathcal {V}}$. The stabilizer of $v_{\mathfrak {k}}\in |{\mathcal {V}}|= V_{\overline{\mathbf {0}}}$ is a complex Lie supergroup $({\mathcal {K}},\mathcal {O}_{\mathcal {K}})$ such that $\mathrm {Lie}({\mathcal {K}})=\mathfrak {k}$.

Proposition C.4

The orbit map of $v_{\mathfrak {k}}$ factors through an embedding $(\mathsf {p}_{v_{\mathfrak {k}}},\mathsf {p}_{v_{\mathfrak {k}}}^\#):{\mathcal {G}}/{\mathcal {K}}\hookrightarrow {\mathcal {V}}$ whose image is an open subsupermanifold of ${\mathcal {V}}$.

Proof

This follows from the fact that the differential of the orbit map $(\mathsf {p}_{v_{\mathfrak {k}}},\mathsf {p}_{v_{\mathfrak {k}}}^\#)$ is a bijection for all $g\in {\mathcal {G}}$, which is a consequence of Lemma C.3 and ${\mathcal {G}}$-equivariance of $(\mathsf {p}_{v_{\mathfrak {k}}},\mathsf {p}_{v_{\mathfrak {k}}}^\#)$. $\square$

Remark C.5

Using the embedding ${\mathcal {G}}/{\mathcal {K}}\hookrightarrow {\mathcal {V}}$ of Proposition C.4 and the natural injection $\mathscr {P}(V)\hookrightarrow \mathcal {O}_{{\mathcal {V}}}(|{\mathcal {V}}|)$, we obtain a ${\mathcal {G}}$-equivariant embedding

$$\begin{aligned} \mathsf {p}_{v_{\mathfrak {k}}}^\#(|{\mathcal {V}}|)\big |_{\mathscr {P}(V)}: \mathscr {P}(V)\hookrightarrow \mathcal {O}_{{\mathcal {G}}/{\mathcal {K}}}\left( |{\mathcal {G}}/{\mathcal {K}}|\right) . \end{aligned}$$

Furthermore, connectedness of $|{\mathcal {G}}|$ implies $\mathscr {PD}(V)^{\mathfrak {g}}=\mathscr {PD}(V)^{\mathcal {G}}$. Therefore we can restrict every $D\in \mathscr {PD}(V)^{\mathfrak {g}}$ to the open subsupermanifold ${\mathcal {G}}/{\mathcal {K}}$ of ${\mathcal {V}}$, and indeed $D\big |_{{\mathcal {G}}/{\mathcal {K}}}\in \mathscr {D}({\mathcal {G}}/{\mathcal {K}})$.

For the next proposition, recall that every linear functional $\varphi :\tilde{\mathfrak {h}}\rightarrow {\mathbb {C}}$ induces a natural homomorphism of ${\mathbb {C}}$-algebras $\mathcal {S}\big (\tilde{\mathfrak {h}}\big )\rightarrow {\mathbb {C}}$ given by $x_1\cdots x_k\mapsto \varphi (x_1)\cdots \varphi (x_k)$ for $x_1,\ldots ,x_k\in \tilde{\mathfrak {h}}$. Set $\mathcal {S}^{(d)}(\tilde{\mathfrak {h}}):=\bigoplus _{i=0}^d\mathcal {S}^i\big (\tilde{\mathfrak {h}}\big )$.

Proposition C.6

Assume that $\mathscr {P}(V)$ is a completely reducible and multiplicity-free $\mathfrak {g}$-module, and let $D\in \mathscr {PD}^{(d)}(V)^{\mathfrak {g}}$. Then there exists an element $x_D\in \mathcal {S}^{(d)}\big (\tilde{\mathfrak {h}}\big )$ such that for every irreducible $\mathfrak {g}$-module $W\subseteq \mathscr {P}(V)$, the action of D on W is by the scalar ${\tilde{\lambda }}(x_D)$, where ${\tilde{\lambda }}$ is the $\tilde{\mathfrak {b}}$-highest weight of W.

Proof

Let ${\mathcal {G}}$, ${\mathcal {V}}$, and ${\mathcal {K}}$ be defined as above, and let $(\mathsf {q},\mathsf {q}^\#):{\mathcal {G}}\rightarrow {\mathcal {G}}/{\mathcal {K}}$ be the canonical quotient map. Then $\mathsf {q}^\#(|{\mathcal {G}}/{\mathcal {K}}|):\mathcal {O}_{{\mathcal {G}}/{\mathcal {K}}}(|{\mathcal {G}}/{\mathcal {K}}|)\rightarrow \mathcal {O}_{{\mathcal {G}}}(G)$ is an injection. For any $\tilde{X}\in \mathbf {U}(\mathfrak {g})$, let $\mathrm {L}_{{\tilde{X}}}$ (respectively, $\mathrm {R}_{{\tilde{X}}}$) denote the action of ${\tilde{X}}$ on $\mathcal {O}_{\mathcal {G}}(G)$ by left invariant (respectively, right invariant) differential operators. By Proposition C.1, there exists ${\tilde{D}}\in \mathbf {U}^{(d)}(\mathfrak {g})^{{\mathcal {K}}}$ such that $\mathsf {q}^\#(|{\mathcal {G}}/{\mathcal {K}}|)(Df)= \mathrm {L}_{{\tilde{D}}} \mathsf {q}^\#(|{\mathcal {G}}/{\mathcal {K}}|)(f)$ for every $f\in \mathcal {O}_{{\mathcal {G}}/{\mathcal {K}}}(|{\mathcal {G}}/{\mathcal {K}}|)$.

Now set $f:=\mathsf {p}_{v_{\mathfrak {k}}}^\#(|V|)(\phi _{{\tilde{\lambda }}})$ where $\phi _{{\tilde{\lambda }}}\in \mathscr {P}(V)$ is a highest weight vector of W, and let ${\tilde{f}}:=\mathsf {q}^\#(|{\mathcal {G}}/{\mathcal {K}}|)(f)$. Let ${\mathcal {N}}$ be the connected Lie subsupergroup of ${\mathcal {G}}$ such that $\mathrm {Lie}({\mathcal {N}})=\tilde{\mathfrak {n}}$. Then ${\tilde{f}}$ is left ${\mathcal {N}}$-invariant and right ${\mathcal {K}}$-invariant. We can express ${\tilde{D}}$ in the form ${\tilde{D}}=D_1+D_2+D_3$, where

$$\begin{aligned} D_1\in \tilde{\mathfrak {n}}\mathbf {U}^{(d-1)}(\mathfrak {g}), \ D_2\in \mathbf {U}^{(d)}\big (\tilde{\mathfrak {h}}\big ), \text { and } D_3\in \mathbf {U}^{(d-1)}(\mathfrak {g})\mathfrak {k}. \end{aligned}$$

From ${\mathcal {K}}$-invariance of f it follows that $\mathrm {L}_{D_3}{\tilde{f}}=0$. Furthermore, we can write $D_1$ as a sum of elements of the form $XD'$ where $X\in \tilde{\mathfrak {n}}$ and $D'\in \mathbf {U}(\mathfrak {g})$. Let ${\mathcal {H}}:=(H,\mathcal {O}_{{\mathcal {H}}})$ denote the connected Lie subsupergroup of ${\mathcal {G}}$ such that $\mathrm {Lie}({\mathcal {H}})=\tilde{\mathfrak {h}}$. For $h\in H$ we have

$$\begin{aligned} \mathrm {L}_{XD'}{\tilde{f}}(h) =\mathrm {L}_X\left( \mathrm {L}_{D'}{\tilde{f}} \right) (h)= \mathrm {R}_{-\mathrm {Ad}_hX} \left( \mathrm {L}_{D'}{\tilde{f}} \right) (h). \end{aligned}$$

Since $\mathrm {L}_{D'}{\tilde{f}}$ is left ${\mathcal {N}}$-invariant and $\mathrm {Ad}_h\left( \tilde{\mathfrak {n}}\right) \subseteq \tilde{\mathfrak {n}}$, it follows that $\mathrm {L}_{XD'}{\tilde{f}}(h)=0$. Consequently, we have shown that for $x_D:=D_2$,

$$\begin{aligned} \mathrm {L}_{{\tilde{D}}}{\tilde{f}}(h)=\mathrm {L}_{D_2}{\tilde{f}}(h)=\tilde{\lambda }\big (D_2\big ){\tilde{f}}(h) ={\tilde{\lambda }}(x_D){\tilde{f}}(h) . \end{aligned}$$

It remains to prove that ${\tilde{f}}\ne 0$. To this end, note that the canonical multiplication morphism

$$\begin{aligned} {\mathcal {N}}\times {\mathcal {H}}\times {\mathcal {K}}\rightarrow {\mathcal {G}}\end{aligned}$$

is a local isomorphism at the identity element. Thus, from analyticity, left ${\mathcal {N}}$-invariance, and right ${\mathcal {K}}$-invariance of ${\tilde{f}}$, it follows that ${\tilde{f}}\big |_{H}$ is not identically zero. $\square$