Selfextensions de modules de Harish-Chandra et une question de I. M. Gelfand. (French) Zbl 0559.17007

Delorme, Patrick

Homologie, groupes Ext\(^ n\) représentations de longueur finie des groupes de Lie, Astérisque 124/125, 31-48 (1985).

[For the entire collection see Zbl 0547.00015.]
Let G be a semisimple group, and X an irreducible Harish-Chandra module for G. This paper is concerned with the possible self-extensions of X. It is shown that they are in some sense controlled by the self-extensions of a one-dimensional module over a nice commutative algebra (usually a polynomial algebra). For X a generic principal series for a complex group, this control is made into an explicit equivalence of categories, verifying a conjecture of Gelfand.

Reviewer: D.Vogan

MSC:

17B55	Homological methods in Lie (super)algebras
22E47	Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)

Keywords:

semisimple group; irreducible Harish-Chandra module; self-extensions; complex group; conjecture of Gelfand