Contemporary Mathematics

Volume 1114, 1993

Introduction to Representations

in Analytic Cohomology

A. W. KNAPP

ABSTRACT.

This is a survey of background and old results concerning rep-

resentations in cohomology sections of vector bundles. The base space is a

homogeneous space

G/L,

where

G

is a connected reductive Lie group and

L

is the centralizer of a torus. When

G

is compact, the representations

in question are the subject of the Bott-Borel-Weil Theorem. When G is

noncompact and L is compact, the representations are identified by the

Langlands Conjecture, which was proved by Schmid. For noncompact L,

difficult analytic problems blocked progress initially. To avoid these difficul-

ties, Zuckerman and Vogan developed an algebraic analog, cohomological

induction, that gave a construction of identifiable representations that were

often irreducible unitary. Recent progress has related the analytic repre-

sentations and their algebraic analogs in various ways.

1.

Sections of homogeneous vector bundles

This paper gives some background from representation theory for understand-

ing the connection between the Penrose transform and analytic realizations of

group representations.

It

is assumed that the reader is acquainted with elemen-

tary facts about holomorphic vector bundles and the elementary structure theory

of semisimple groups. Discussions of these two topics may be found in Wells (19,

Chapter I] and Knapp (8, Chapter V], respectively. The results in this paper

largely are not new, and, for the most part, references will be given in place of

proofs.

1991 Mathematics Subject Classification. Primary 20G05, 22E45, 32L10; Secondary 55R91,

83C60.

The author was supported by NSF Grant DMS 91 00367.

This paper is in final form and no version of it will be submitted for publication elsewhere.

1

©

1993 American Mathematical Society

0271-4132/93 $1.00

+

$.25 per page

http://dx.doi.org/10.1090/conm/154/01353