Contemporary Mathematics
Volume 1114, 1993
Introduction to Representations
in Analytic Cohomology
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
is a connected reductive Lie group and
is the centralizer of a torus. When
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.
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.
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
1991 Mathematics Subject Classification. Primary 20G05, 22E45, 32L10; Secondary 55R91,
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.
1993 American Mathematical Society
0271-4132/93 $1.00
$.25 per page
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