Schur $Q$-functions and the Capelli eigenvalue problem for the Lie superalgebra $\mathfrak{q}(n)$

Abstract

Let $\mathfrak{l}:=\mathfrak{q}(n)\times \mathfrak{q}(n)$, where $\mathfrak{q}(n)$ denotes the queer Lie superalgebra. The associative superalgebra $V$ of type $Q(n)$ has a left and right action of $\mathfrak{q}(n)$, and hence is equipped with a canonical $\mathfrak{l}$-module structure. We consider a distinguished basis $\{D_{\lambda }\}$ of the algebra of $\mathfrak{l}$-invariant super-polynomial differential operators on $V$, which is indexed by strict partitions of length at most $n$. We show that the spectrum of the operator $D_{\lambda }$, when it acts on the algebra $\mathcal{P}(V)$ of super-polynomials on $V$, is given by the factorial Schur $Q$-function of Okounkov and Ivanov. This constitutes a refinement and a new proof of a result of Nazarov, who computed the top-degree homogeneous part of the Harish-Chandra image of $D_{\lambda }$. As a further application, we show that the radial projections of the spherical super-polynomials corresponding to the diagonal symmetric pair $(\mathfrak{l},\mathfrak{m})$, where $\mathfrak{m}:=\mathfrak{q}(n)$, of irreducible $\mathfrak{l}$-submodules of $\mathcal{P}(V)$ are the classical Schur $Q$-functions.

To appear in the Contemporary Mathematics volume honoring Gestur Olafsson's 65th birthday

References in corpus (1)

• Spherical representations of Lie supergroups