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December, 2004 Generalized Harish-Chandra Modules with Generic Minimal t-Type
Ivan Penkov, Gregg Zuckerman
Asian J. Math. 8(4): 795-812 (December, 2004).

Abstract

We make a first step towards a classification of simple generalized Harish-Chandra modules which are not Harish-Chandra modules or weight modules of finite type. For an arbitrary algebraic reductive pair of complex Lie algebras(g,t), we construct, via cohomological induction, the fundamental series F . (p,E) of generalized Harish-Chandra modules. We then use F. (p,E) to characterize any simple generalized Harish-Chandra module with generic minimal t-type. More precisely, we prove that any such simple(g,t)-module of finite type arises as the unique simple submodule of an appropriate fundamental series module Fs (p,E) in the middle dimension s. Under the stronger assumption that t contains a semisimple regular element of g, we prove that any simple(g,t)-module with generic minimal t-type is necessarily of finite type, and hence obtain a reconstruction theorem for a class of simple(g,t)-module which can a priori have infinite type. We also obtain generic general versions of some classical theorems of Harish-Chandra, such as the Harish-Chandra admissibility theorem. The paper is concluded by examples, in particular we compute the genericity condition on a ttype for any pair (g,t)with t about equal to sl(2).

Citation

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Ivan Penkov. Gregg Zuckerman. "Generalized Harish-Chandra Modules with Generic Minimal t-Type." Asian J. Math. 8 (4) 795 - 812, December, 2004.

Information

Published: December, 2004
First available in Project Euclid: 13 June 2005

zbMATH: 1079.17004
MathSciNet: MR2127949

Rights: Copyright © 2004 International Press of Boston

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Vol.8 • No. 4 • December, 2004
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