## Pavel Etingof

Representations of Lie Groups

### About the Book

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.

### About the Author

**Pavel Etingof** is a Professor of Mathematics at MIT (since 2005) and the Chief Research Advisor of the MIT PRIMES Program (Program for Research in Mathematics, Engineering and Science for High School Students) since 2010.

Etingof received an M.S. in Applied Mathematics from the Moscow Oil & Gas Institute in 1989, and a Ph.D. in Mathematics from Yale University in 1994. Igor Frenkel was his thesis advisor. He went to Harvard as a Benjamin Peirce Assistant Professor in 1994, and joined the MIT mathematics faculty as Assistant Professor in 1998, and Professor in 2005. Etingof’s research interests are primarily in studies which intersect representation theory and mathematical physics, such as quantum groups. He serves as chief editor of the Journal of the AMS and of Selecta Math. He has co-authored 8 texts. He served as Chair of the Graduate Student Committee from 2002-05. In 1999 Etingof received a Clay Mathematics Institute Prize fellowship. In 2012 he was selected to be the Robert E. Collins Distinguished Scholar in the Mathematics Department. He was named a Fellow of the AMS in the 2013 Inaugural Class. He was selected by the Institute for the Frank E. Perkins Award for excellence in graduate advising in 2015, and again in 2018. In 2016 he was elected Fellow of the American Academy of Arts and Sciences.

### Managing Editors

- Colin Adams
- Eric Friedlander
- Sergei Tabachnikov

### Editorial Board

- George Andrews
- Colin Adams
- Eric Friedlander
- Robert Ghrist
- Joel Hass
- Robion Kirby
- Alex Kontorovich
- Sergei Tabachnikov

### AMR Research Monographs

Volume 5

**ISBN: 978-1-959384-05-2**

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