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A branching law for subgroups fixed by an involution and a noncompact analogue of the Borel-Weil theorem

  • Bertram Kostant
Chapter
Part of the Progress in Mathematics book series (PM, volume 220)

Abstract

Let g be a complex semisimple Lie algebra and let t be the subalgebra of fixed elements in g under the action of an involutory automorphism of g. Any such involution is the complexification of the Cartan involution of a real form of g. If V λ is an irreducible finite-dimensional representation of g, the Iwasawa decomposition implies that V λ is a cyclic U(t)module where the cyclic vector is a suitable highest weight vector v λ. In this paper we explicitly determine generators of the left ideal annihilator L λ (t) of v λ in U(t). One of the applications of this result is a branching law which determines how V λ decomposes as a module for t Other applications include (1) a new structure theorem for the subgroup M (conventional terminology) and its unitary dual, and (2) a generalization of the Cartan-Helgason theorem where, in the generalization, the trivial representation of M (using conventional terminology) is replaced by an arbitrary irreducible representation τ of M. For the generalization we establish the existence of a unique minimal representation of g associated to τ.

Another application (3) yields a noncompact analogue of the Borel-Weil theorem. For a suitable semisimple Lie group G and maximal compact subgroup K, where g = (Lie G) and t = (Lie K) the representation V λ embeds uniquely (see [W], §8.5) as a finite dimensional (g, K) submodule V λ of the Harish-Chandra module H(δ, ξ) of a principal series representation of G. The functions in H(δ, ξ) are uniquely determined by their restriction to K. As a noncompact analogue of the Borel-Wei! theorem the functions in V λ are given as solutions of differential equations arising from the generators of L λ(t) (rather than Cauchy-Riemann equations as in the Borel-Weil theorem).

Keywords

Disjoint Union Left Ideal Cartan Subalgebra Maximal Compact Subgroup Principal Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Bertram Kostant
    • 1
  1. 1.Department of MathematicsMITCambridgeUSA

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