# Workshop on Unitary Representations of Real Reductive Groups

Reading List

We recommend all attendees download and install the atlas software (available for linux, Mac and Windows). Then go to Getting Started (on the documentation web site) to start learning about the software.

**Here is some material everyone should be familiar with before the workshop begins.**

A good place to start is Infinite Dimensional Representations of Real Reductive Groups, an overview by David Vogan.

Also highly recommended are the first few chapters of two books by David Vogan.

*Unitary representations of reductive Lie groups*, volume 118
of *Annals of Mathematics Studies*,
Princeton University Press, Princeton, NJ, 1988 (the orange book).
The Introduction is a good survey. Chapter 1 is a good exposition of the representation theory of
compact groups.
(In a first reading assume the group is connected, and skim the discussion about large and small
Cartan subgroups.)
Read chapter 2 for the passage from the analytic theory of group representations, to the algebraic
one of Harish-Chandra modules.

*Representations of real reductive Lie groups*, volume 15 of
*Progress in Mathematics*
Birkhauser Boston, Mass., 1981 (the green book).
Chapter 0 is a somewhat more formal introduction to the material in the orange book.
We'll be spending a lot of time on SL(2,R), roughly following Chapter 1.

Section 1 of Algorithms for Representation Theory of Real Reductive Groups also has a good introduction, with the Atlas approach in mind. Sections 2 and 3 have important background material about algebra groups, root data, and real groups.

The Langlands Classification is a good survey of the Langlands classification, based on the example of SL(2,R).

Unitary Representations of Real Reductive Groups by Jeffrey Adams, Peter Trapa, Marc van Leeuwen, and David A. Vogan, Jr. has all the details of the algorithm for computing Hermitian forms. Everyone should at least read the Introduction.

**References from the lectures**

Here are some papers which we will be referring to during the lectures.

- Root Systems (Chapter VI/Section 1) from Bourbaki, Groupes et Algebras de Lie, Chapters 4-6 (in French; there is an English edition. Other references include the books by Springer or Borel on algebraic groups).
- Galois and Cartan cohomology of real groups, by Jeffrey Adams and Olivier Taibi. Details of the "σ/θ picture"
- Branching to a Maximal Compact Subgroup, by David Vogan. The background for understanding K-types in Atlas.
- SL(2) refcard: many facts about SL(2,R) on a single page
- Equivalence of atlas Parameters