DISCRETE DECOMPOSABILITY OF THE RESTRICTION OF

A_{𝔮}(λ) WITH RESPECT TO REDUCTIVE SUBGROUPS III

— restriction of Harish-Chandra modules and associated varieties

Toshiyuki KOBAYASHI

University of Tokyo

Let H ⊂ G 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 π = \overline{A_{𝔮}(λ)}, and proves that the sufficient condition established by T. Kobayashi, [Invent. Math. 117, 181-205 (1994; Zbl. 826.22015)] is in fact necessary. As a corollary of the proof, a finite multiplicity theorem in the branching law is also proved for discretely decomposable modules with respect to a semisimple symmetric pair, which is in sharp contrast to known counter examples regarding the continuous spectrum. As an application, the restriction π|_{H} of a discrete series representation π for a semisimple symmetric space G/H is also discussed. Partial results were announced in [Proc. Japan Acad. Ser. A. 71, 24–26 (1995; Zbl. 828.22030)].

Graduate School of Mathematical Sciences, University of Tokyo,

Meguro, Komaba, 153, Tokyo, Japan

E-mail address: toshi@ms.u-tokyo.ac.jp

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 22E46; Secondary 22E47, 43A85, 22E15..

Key words and phrases. Vogan-Zuckerman derived functor modules, restriction of unitary representations, associated varieties, discrete series representations for semisimple symmetric spaces, finite multiplicity, branching laws.