Journal of the Mathematical Society of Japan

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


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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

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

First available in Project Euclid: 1 August 2011

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Zentralblatt MATH identifier

Primary: 22E47: Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) [See also 17B10]
Secondary: 17B45: Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx]

Hermitian symmetric space KC-invariant differential operator principal symbol Capelli identity generating function twisted moment map


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.

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