Geometric constructions of representations. (English) Zbl 0706.22012

Schmid, Wilfried

Representations of Lie groups: analysis on homogeneous spaces and representations of Lie groups, Proc. Symp., Kyoto/Jap. and Hiroshima/Jap. 1986, Adv. Stud. Pure Math. 14, 349-368 (1988).

[For the entire collection see Zbl 0694.00014.]
From the author’s introduction: The paper describes recent results by J. A. Wolf and the author, on the orbit method for semisimple coadjoint orbits of semisimple Lie groups, without restriction on the type of polarization. Properly interpreted (to avoid analytic difficulties), a procedure suggested by B. Kostant in 1965 actually yields global representations of the group on Fréchet spaces, given as cohomology groups (in several, canonically isomorphic, forms). Moreover, the underlying Harish-Chandra modules coincide with the derived functor modules which correspond to the same data.
The paper contains no detailed proofs (which are to appear elsewhere), but a discussion of the main ideas and related methods and objects: canonical globalizations of Harish-Chandra modules, Zuckerman’s derived functor construction, \({\mathcal D}\)-module construction of Beilinson- Bernstein, vanishing cohomology theorem...

Reviewer: F.Rouvière

MSC:

22E46	Semisimple Lie groups and their representations
57T15	Homology and cohomology of homogeneous spaces of Lie groups

Keywords:

orbit method; semisimple coadjoint orbits; semisimple Lie groups; global representations; cohomology groups; Harish-Chandra modules; derived functor construction