## Journal of the Mathematical Society of Japan

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

Takashi HASHIMOTO

#### Abstract

Let $(G,K)$ be one of the following Hermitian symmetric pair: $SU(p,q),S(U(p) \: \times \: U(q)))$, $(Sp(n,$\mathbf{R}$,U(n))$, or $(SO^*(2n),U(n))$. Let $G_C$ and $K_C$ be the complexifications of $G$ and $K$, respectively, $Q$ the maximal parabolic subgroup of $G_C$ whose Levi part is $K_C$, and $V$ the holomorphic tangent space at the origin of $G/K$. It is known that the ring of $K_C$ -invariant differential operators on $V$ has a generating system $\{\mathit{\Gamma_k}\}$ 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 $G_C / Q$ provides a generating function for the principal symbols of $\mathit{\Gamma_k}$'s.

#### Article information

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

Dates
First available in Project Euclid: 1 August 2011

http://projecteuclid.org/euclid.jmsj/1312203803

Digital Object Identifier
doi:10.2969/jmsj/06330837

Mathematical Reviews number (MathSciNet)
MR2836747

Zentralblatt MATH identifier
1226.22019

#### Citation

HASHIMOTO, Takashi. On the principal symbols of $K_C$- invariant differential operators on Hermitian symmetric spaces. J. Math. Soc. Japan 63 (2011), no. 3, 837--869. doi:10.2969/jmsj/06330837. http://projecteuclid.org/euclid.jmsj/1312203803.

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