Generalized Harish-Chandra modules with generic minimal $\frak k$-type
| dc.creator | Penkov, Ivan | |
| dc.creator | Zuckerman, Gregg | |
| dc.date | 2004-09-16 | |
| dc.date.accessioned | 2026-07-07T05:12:13Z | |
| dc.date.available | 2026-07-07T05:12:13Z | |
| dc.description | 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 $F^\cdot (\p,E)$ of generalized Harish-Chandra modules. We then use $F^\cdot (\p,E)$ 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 $\k\simeq s\ell (2)$. | |
| dc.description | 26 pages | |
| dc.identifier | https://arxiv.org/abs/math/0409285 | |
| dc.identifier | http://arxiv.org/abs/math/0409285 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72500 | |
| dc.subject | Representation Theory | |
| dc.subject | 17B10 | |
| dc.title | Generalized Harish-Chandra modules with generic minimal $\frak k$-type | |
| dc.type | text |