A note on the Capelli identities for symmetric pairs of Hermitian type

TL;DR

This paper introduces new Capelli identities for symmetric pairs of Hermitian type, expressing invariant differential operators in determinantal form and connecting to classical determinant formulas.

Contribution

It presents novel non-commutative Capelli identities for symmetric pairs, extending classical determinant formulas to invariant differential operators in Hermitian symmetric spaces.

Findings

Identities expressed in determinantal form

Reduction to classical formulas via principal symbols

Connection to invariant differential operators in symmetric spaces

Abstract

We get several identities of differential operators in determinantal form. These identities are non-commutative versions of the formula of Cauchy-Binet or Laplace expansions of determinants, and if we take principal symbols, they are reduced to such classical formulas. These identities are naturally arising from the generators of the rings of invariant differential operators over symmetric spaces, and have strong resemblance to the classical Capelli identities. Thus we call those identities the Capelli identities for symmetric pairs.