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 $(SO*(2n), U(n))$. 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.