Year: 2004
Type: article
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, k), 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 k-type.More precisely, we prove that any such simple (g, k)-module of finite type arises as the unique simple submodule of an appropriate fundamental series module F s (p, E) in the middle dimension s.Under the stronger assumption that k contains a semisimple regular element of g, we prove that any simple (g, k)-module with generic minimal k-type is necessarily of finite type, and hence obtain a reconstruction theorem for a class of simple (g, k)-modules 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 k-type for any pair (g, k) with k ≃ sℓ(2).
Source: Asian Journal of Mathematics
Authors: Ivan Penkov, Gregg Zuckerman
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1998579221