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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Mathematical Reviews number (MathSciNet)

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. Journal of the Mathematical Society of Japan 63 (2011), no. 3, 837--869. doi:10.2969/jmsj/06330837.

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  • T. Hashimoto, A central element in the universal enveloping algebra of type $\mathsf{D}_n$ via minor summation formula of Pfaffians, J. Lie Theory, 18 (2008), 581–594.
  • T. Hashimoto, Generating function for $\mathrm{GL}_{n}$-invariant differential operators in the skewCapelli identity, 2009, Lett. Math. Rhys., 93 (2010), 157–168, arXiv:0803.1339v2 [math.RT].
  • T. Hashimoto, K. Ogura, K. Okamoto and R. Sawae, Borel-Weil theory and Feynman path integrals on flag manifolds, Hiroshima Math. J., 23 (1993), 231–247.
  • R. Howe and T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann., 290 (1991), 565–619.
  • M. Ishikawa and M. Wakayama, Application of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities, J. Comb. Theory A, 113 (2006), 113–155.
  • M. Itoh, A Cayley-Hamilton theorem for the skew Capelli elements, J. Algebra., 242 (2001), 740–761.
  • M. Itoh and T. Umeda, On central elements in the universal enveloping algebras of the orthogonal Lie algebra, Compositio Math., 127 (2001), 333–359.
  • K. Kinoshita and M. Wakayama, Explicit Capelli identities for skew symmetric matrices, Proc. Edinburgh Math. Soc., 45 (2002), 449–465.
  • A. W. Knapp, Representation theory of semisimple groups: An overview based on examples, Princeton Mathematical Series, 36, Princeton Univ. Press, 1986.
  • D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory (3rd enlarged edition), Springer-Verlag, 1994.
  • W. Schmid and K. Vilonen, Characteristic cycles of constructible sheaves, Invent. Math., 124 (1996), 451–502.
  • G. Shimura, On differential operators attached to certain representations of classical groups, Invent. Math., 77 (1984), 463–488.