Quadratic Capelli operators and Okounkov polynomials

TL;DR

This paper introduces quadratic Capelli operators on Grassmannians, showing their eigenvalues are given by Okounkov interpolation polynomials, thus linking invariant differential operators with special functions in representation theory.

Contribution

It constructs a new family of invariant differential operators called quadratic Capelli operators and establishes their eigenvalues relate to Okounkov polynomials, extending Capelli theory.

Findings

Eigenvalues of quadratic Capelli operators are given by Okounkov interpolation polynomials.

Construction of invariant differential operators on Grassmannians using double fibration.

Connects Capelli operators with Okounkov polynomials in representation theory.

Abstract

Let $Z$ be the symmetric cone of $r \times r$ positive definite Hermitian matrices over a real division algebra $\mathbb F$. Then $Z$ admits a natural family of invariant differential operators -- the Capelli operators $C_\lambda$ -- indexed by partitions $\lambda$ of length at most $r$, whose eigenvalues are given by specialization of Knop--Sahi interpolation polynomials. In this paper we consider a double fibration $Y \longleftarrow X \longrightarrow Z$ where $Y$ is the Grassmanian of $r$-dimensional subspaces of $\mathbb F^n $ with $n \geq 2r$. Using this we construct a family of invariant differential operators $D_{\lambda,s}$ on $Y$ that we refer to as quadratic Capelli operators. Our main result shows that the eigenvalues of the $D_{\lambda,s}$ are given by specializations of Okounkov interpolation polynomials.