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