The Penrose transform for Sp(n, R) and singular unitary representations

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概要

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• We give a general definition of the Radon-Penrose transform for a Zuckerman-Vogan derived functor module of a reductive Lie group G, which maps from the Dolbeault cohomology group over a pseudo-Kähler homogeneous manifold into the space of smooth sections of a vector bundle over a Riemannian symmetric space. Furthermore, we formulate a functorial property between two Penrose trans-forms in the context of the Kobayashi theory of discretely decomposable restrictions of unitary representations.Based on this general theory, we study the Penrose transform for a family of sin-gular unitary representations of Sp(n, \bm{R}) in details. We prove that the image of the Penrose transform is exactly the space of global holomorphic solutions of the system of partial differential equations of minor determinant type of odd degree over the bounded symmetric domain of type CI, which is biholomorphic to the Siegel upper half space. This system might be regarded as a generalization of the Gauss-Aomoto-Gelfand hy-pergeometric differential equations to higher order. We also find a new phenomenon that the kernel of the Penrose transform is non-zero, which we determine explicitly by means of representation theory.

• 社団法人 日本数学会の論文

著者

• Sekiguchi Hideko

  Department Of Mathematical Sciences University Of Tokyo

• Sekiguchi Hideko

  Department Of Mathematics Kobe University

関連論文

• Commuting families of differential operators invariant under the action of a Weyl group
• Combinatorial formula of the dimension of global solutions to a generalized hypergeometric system Μ_(ν)
• The Penrose transform for Sp(n, R) and singular unitary representations
• The Penrose Transform for Certain Non-Compact Homogeneous Manifolds of $U(n,n)$

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