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

2008/04/25 by Takashi Hashimoto, Hashimoto, Takashi
Mathematics · #17B45 #22E47 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B45 #msc:22E47

paper · pdf · doi:10.48550/arxiv.0804.4038

23pages, no figure

arxiv created 2008/04/25 · arxiv updated 2009/12/01

Abstract

Let (G,K) be one of the following classical irreducible Hermitian symmetric pairs of noncompact type: (SU(p,q), S(U(p) × 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.

Related

• A note on the Capelli identities for symmetric pairs of Hermitian type
• Complexified Hermitian Symmetric Spaces, Hyperkähler Structures, and Real Group Actions
• Degenerate principal series representations and their holomorphic extensions
• Generating function for GL_n-invariant differential operators in the skew Capelli identity
• Complex symmetry of first-order differential operators on Hardy space