Quadratic Capelli operators and Okounkov polynomials

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 $C_λ$ -- indexed by partitions $λ$ 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 $D_{λ,s}$ on $Y$ that we refer to as quadratic Capelli operators. Our main result shows that the eigenvalues of the $D_{λ,s}$ are given by specializations of Okounkov interpolation polynomials.