### On the principal symbols of KC-invariant differential operators on Hermitian symmetric spaces

Takashi HASHIMOTO
Source: J. Math. Soc. Japan Volume 63, Number 3 (2011), 837-869.

#### Abstract

Let (G,K) be one of the following Hermitian symmetric pair: (SU(p,q), S(U(p) × U(q))), (Sp(n,R), U(n)), or (SO*(2n), U(n)). Let Gc and KC be the complexifications of G and K, respectively, Q the maximal parabolic subgroup of Gc whose Levi part is KC, and V the holomorphic tangent space at the origin of G/K. It is known that the ring of KC-invariant differential operators on V has a generating system {Γk} given in terms of determinant or Pfaffian that plays an essential role in the Capelli identities. Our main result is that determinant or Pfaffian of a deformation of the twisted moment map on the holomorphic cotangent bundle of Gc/Q provides a generating function for the principal symbols of Γk's.

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Primary Subjects: 22E47
Secondary Subjects: 17B45
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Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1312203803
Digital Object Identifier: doi:10.2969/jmsj/06330837
Mathematical Reviews number (MathSciNet): MR2836747

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