On the principal symbols of $K_{\mathbb{C}}$-invariant differential operators 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 $(SO\ast (2n),U(n))$. 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 $\{{\Gamma }_k\}$ 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 ${\Gamma }_k$'s.

23pages, no figure

References in corpus (1)

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

Cited by in corpus (1)

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