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

Abstract

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 = Aq(λ), 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.