On the principal symbols of $K_{\mathbb C}$-invariant differential operators on Hermitian symmetric spaces

TL;DR

This paper investigates the principal symbols of $K_{ ext{C}}$-invariant differential operators on Hermitian symmetric spaces, revealing that determinants or Pfaffians of the moment map generate these symbols, connecting to Capelli identities.

Contribution

It establishes that the principal symbols of invariant differential operators are generated by determinants or Pfaffians of the moment map, linking geometric and algebraic structures.

Findings

Principal symbols generated by determinants or Pfaffians of the moment map.

Connection established between principal symbols and Capelli identities.

Provides a generating function for principal symbols on Hermitian symmetric spaces.

Abstract

Let $(G,K)$ be one of the following classical irreducible Hermitian symmetric pairs of noncompact type: $(SU(p,q), S(U(p) \times U(q))),(Sp(n,R), U(n))$, or $(SO*(2n), U(n))$. Let $G_{\mathbb C}$ and $K_{\mathbb C}$ be complexifications of $G$ and $K$, respectively, and let $P$ be a maximal parabolic subgroup of $G_{\mathbb C}$ whose Levi subgroup is $K_{\mathbb C}$. Let $V$ be the holomorphic part of the complexifiaction of the tangent space at the origin of $G/K$. It is well known that the ring of $K_{\mathbb C}$-invariant differential operators on $V$ has a generating system $\{\varGamma_k \}$ given in terms of determinant or Pfaffian that plays an essential role in the Capelli identities. Our main result of this paper is that determinant or Pfaffian of the ``moment map'' on the holomorphic cotangent bundle of $G_{\mathbb C}/P$ provides a generating function for the principal symbols of $\varGamma_k$'s.