Generating function for GL_n-invariant differential operators in the skew Capelli identity

TL;DR

This paper constructs a generating function for GL_n-invariant differential operators related to the skew Capelli identity, using Pfaffians of matrices with operators and Hermite polynomial coefficients.

Contribution

It introduces a novel generating function for invariant differential operators in the context of the skew Capelli identity, linking Pfaffians, Hermite polynomials, and group actions.

Findings

Pfaffian of a matrix with operators generates GL_n-invariant differential operators

The coefficients of the generating function are Hermite polynomials

Provides a new algebraic tool for understanding the skew Capelli identity

Abstract

Let Alt_n be the vector space of all alternating n-by-n complex matrices, on which the complex general linear group GL_n acts by $x \mapsto g x g^t$. The aim of this paper is to show that Pfaffian of a certain matrix whose entries are multiplication operators or derivations acting on polynomials on Alt_n provides a generating function for the GL_n-invariant differential operators that play a role in the skew Capelli identity, with coefficients the Hermite polynomials.