On the Representation Theory of Semisimple Lie Groups

Lie groups;representation theory;geometry;Pure Mathematics

Al-Faisal, Faisal

University of Waterloo

关键词: Lie groups;    representation theory;    geometry;    Pure Mathematics;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/5421/1/alfaisal_thesis.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

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.