Inventiones mathematicae

, Volume 131, Issue 2, pp 229–256 | Cite as

Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups III. Restriction of Harish-Chandra modules and associated varieties

III. Restriction of Harish-Chandra modules and associated varieties
  • Toshiyuki Kobayashi


Let HG be real reductive Lie groups and π an irreducible unitary representation of G. We introduce an algebraic formulation (discretely decomposable restriction) to single out the nice class of the branching problem (breaking symmetry in physics) in the sense that there is no continuous spectrum in the irreducible decomposition of the restriction π| H . This paper offers basic algebraic properties of discretely decomposable restrictions, especially for a reductive symmetric pair (G,H) and for the Zuckerman-Vogan derived functor module \(\), and proves that the sufficient condition [Invent. Math. '94] is in fact necessary. A finite multiplicity theorem is established for discretely decomposable modules which is in sharp contrast to known examples of the continuous spectrum. An application to the restriction π| H of discrete series π for a symmetric space G/H is also given.


Symmetric Space Continuous Spectrum Unitary Representation Discrete Series Irreducible Unitary Representation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Toshiyuki Kobayashi
    • 1
  1. 1.Department of Mathematical Sciences, University of Tokyo, Meguro, Komaba, 153, Tokyo, Japan (e-mail:

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