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

## TL;DR

This paper links Schur Q-functions to the spectra of invariant differential operators in the queer Lie superalgebra setting, providing new proofs and insights into the Capelli eigenvalue problem.

## Contribution

It introduces a basis of invariant super-polynomial differential operators related to strict partitions and connects their spectra to factorial Schur Q-functions, offering a new proof of Nazarov's result.

## Key findings

- Spectrum of operators D_λ given by factorial Schur Q-functions
- Refinement and new proof of Nazarov's result on Harish-Chandra images
- Radial projections of super-polynomials are classical Schur Q-functions

## 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 $\mathscr P(V)$ 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 $\mathscr P(V)$ are the classical Schur $Q$-functions.