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On the Representation Theory of Semisimple Lie Groups

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alfaisal_thesis.pdf (694.85 KB)

Date

2010-08-30T20:45:26Z

Authors

Al-Faisal, Faisal

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University of Waterloo

Abstract

This thesis is an expository account of three central theorems in the representation theory of semisimple Lie groups, namely the theorems of Borel-Weil-Bott, Casselman-Osborne and Kostant. The first of these realizes all the irreducible holomorphic representations of a complex semisimple Lie group G in the cohomology of certain sheaves of equivariant line bundles over the flag variety of G. The latter two theorems describe the Lie algebra cohomology of a maximal nilpotent subalgebra of Lie(G) with coefficients in an irreducible Lie(G)-module. Applications to geometry and representation theory are given. Also included is a brief overview of Schmid's far-reaching generalization of the Borel--Weil--Bott theorem to the setting of unitary representations of real semisimple Lie groups on (possibly infinite-dimensional) Hilbert spaces.

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Keywords

Lie groups, representation theory, geometry

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http://hdl.handle.net/10012/5421

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Pure Mathematics