Computational Theory of Real Reductive Groups
A workshop, July 20-24, 2009, at the University of Utah





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Suggestions for Wednesday Afternoon

Annotated Background Reading List:

In preparation for this summer's workshop, we put together the following list of suggested background reading. When possible we tried to find references which were freely available online.

  • Based root data. The starting point for much of the computational theory discussed in the workshop is the classification of connected reductive complex algebraic groups in terms of root data. Some familiarity with these ideas is really prerequisite to most of the lectures.

    A good reference for this material is T.A. Springer's Linear Algebraic Groups (second edition) published in 1998 by Birkhauser in their Progress in Math series, particularly Chapters 7-10. An (incomplete) google preview is available. A short account (available freely online) of some of the main ideas is available from Yiannis Sakellaridis.

    A treatment of based root data in the (equivalent) setting of connected compact Lie groups is contained in Section 2 of the expository paper Three-dimensional subgroups and unitary representations by David Vogan. Ideally this should be read in conjunction with Springer's account.

    A nice self-contained exposition (a la Bourbaki) of the abstract theory of root systems is in Section 5.1 (beginning on page 139) of Dragan Milicic's notes on Lie groups.

  • The Bruhat decomposition. Another essential prerequisite is familiarity with the Bruhat decomposition. This is Section 8.3 in Springer's book mentioned above. (We had trouble locating a complete online reference. Let us know if you find one.)

  • Background on Harish-Chandra modules. Some familiarity with Harish-Chandra modules, as well as an acquaintance with the representation theory of SL(2,R), would be very helpful. Chapter 0 (particularly Section 3) and Chapter 1 of David Vogan's green book Representations of Real Reductive Lie Groups (published in 1981 by Birkhauser) are good places to start. The first twenty pages of the expository paper Representations of semisimple Lie groups by Tony Knapp and Peter Trapa also contains a quick overview of the study of Harish-Chandra modules.

  • Background on Kazhdan-Lusztig theory. As preparation for the lectures of Stembridge and Trapa, it would be useful to have taken at least a peek at Kazhdan and Lusztig's orginal paper "Representations of Coxeter groups and Hecke algebras", Inv. Math, vol 53, number (2), 165-184. A reasonable account exists on wikipedia.