Representations of Lie Groups

This book is based on a one-semester graduate course on representations of non-compact Lie groups given by the author at MIT (and contains a bit more material than fits into a course). It is organized into 31 sections, roughly corresponding to 1.5-hour lectures. The book first covers the basic analytic background (representations on Fr\’echet spaces, algebra of compactly supported measures on the group, smooth and analytic vectors, admissible representations, Harish-Chandra analyticity theorem, etc.), reducing the study of representations to Harish-Chandra modules. The rest is devoted to the algebraic study of Harish-Chandra modules, focusing on Harish-Chandra bimodules (corresponding to complex Lie groups regarded as real groups). This includes the category O, Chevalley restriction theorem, Chevalley-Shepard-Todd theorem, Harish-Chandra isomorphism, theorems of Kostant and Duflo-Joseph, projective functors, equivalence between Harish-Chandra bimodules and category O, classification of irreducible Harish-Chandra bimodules. At the end the book discusses a geometric approach to the subject using K-equivariant D-modules on the flag variety G/B, and derive the classification of irreducible Harish-Chandra modules (due to Langlands) in terms of K-orbits on G/B. Also given is a complete treatment of the representation theory of SL(2,R) and SL(2,C), including unitary representations.