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# Asian Journal of Mathematics

## Volume 8 (2004)

### Number 4

### Generalized Harish-Chandra Modules with Generic Minimal **t**-Type

Pages: 795 – 812

DOI: http://dx.doi.org/10.4310/AJM.2004.v8.n4.a25

#### Authors

#### 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 *F*^{s} (**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 **t**type for any pair (**g**,**t**)with **t** about equal to *sl*(2).