Representations of Lie Groups

Fall 2023

The goal of this course is to give an introduction to the representation theory of compact and non-compact Lie groups. It will rely on some material from {{% resource_link "840f5da9-f923-4ff4-a62e-e705e95de700" "*18.745 Lie Groups and Lie Algebras I*" %}} and {{% resource_link "91493fd6-b586-4a3c-a08d-7d770c40094c" "*18.755 Lie Groups and Lie Algebras II*" %}}, but full familiarity with this material is not required. Topics include continuous representations; algebras of measures on a Lie group; K-finite, smooth, and analytic vectors; admissible representations; unitary representations; the Harish-Chandra admissibility and analyticity theorems; (g,K)-modules; infinitesimal equivalence and realizations; Harish-Chandra’s globalization theorem; representations of SL(2,R), the Chevalley restriction theorem; the Chevalley-Shepard-Todd theorem; Kostant’s theorem; Harish-Chandra isomorphism; Bernstein-Gelfand-Gelfand category O; projective functors; classification of irreducible Harish-Chandra bimodules; the nilpotent cone; the Duflo-Joseph theorem; Duflo’s primitive ideal theorem; the Borel-Weil theorem; Springer resolution; Beilinson-Bernstein localization; D-modules; and classification of representations of a semisimple real group in terms of the orbits of the complexified maximal compact subgroup on the flag variety.