2004/09/16 by Ivan Penkov, Penkov, Ivan, Gregg J. Zuckerman +2
Mathematics · #17B10 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B10
paper · pdf · doi:10.48550/arxiv.math/0409285
26 pages
arxiv created 2004/09/16 · openalex publication_date 2004/09/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Fs(\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).