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 ${\mathfrak{g}}_{♭}\cong {\mathfrak{g}}_{♭}(-1)\oplus {\mathfrak{g}}_{♭}(0)\oplus {\mathfrak{g}}_{♭}(1)$, such that ${\mathfrak{g}}_{♭}(-1)\cong J$. Set $V:={\mathfrak{g}}_{♭}(-1{)}^{\ast }$ and $\mathfrak{g}:={\mathfrak{g}}_{♭}(0)$. In most cases, the space $\mathcal{P}(V)$ of superpolynomials on $V$ is a completely reducible and multiplicity-free representation of $\mathfrak{g}$, with a decomposition $\mathcal{P}(V):={⨁}_{\lambda \in \Omega }{V}_{\lambda }$, 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 $\mathcal{PD}(V{)}^{\mathfrak{g}}$. In this paper we complete the solution to the Capelli eigenvalue problem, which is to determine the scalar $c_{\mu }(\lambda )$ by which $D_{\mu }$ acts on $V_{\lambda }$. We associate a restricted root system $\Sigma$ 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 $\Sigma$ for $\mathcal{P}(V)$ to be completely reducible and multiplicity-free. When $\Sigma$ 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 $\Sigma$ is of type $\mathsf{A}(m,n)$, and Okounkov-Ivanov's factorial Schur $Q$-polynomials when $\Sigma$ is of type $\mathsf{Q}(n)$.