Computing with real reductive groups
 Jeffrey Adams (Maryland) with the assistance of Fokko du Cloux (Lyon)

The representation theory of a real reductive group G, such as GL(n,R), is a very technical subject. The representations themselves are infinite-dimensional, and difficult to write down explicitly. Even the parametrization of the irreducible representations of G is complicated, and difficult for everyone but the cognoscenti. This parametrization known as the Langlands, or Langlands/Knapp/Zuckerman classification. An independent classification is due to Vogan.

The purpose of these talks is to describe a concrete, explicitly computable, combinatorial description of G^. This has been implemented on computer by Fokko du Cloux. I will describe the algorithm itself, with some of the mathematical background. A separate session may be devoted to demonstrating the software.

Here is a little more detail about what the algorithm and software do. Suppose G is a connected, reductive, complex algebraic group, G(R) is a real form of G. Let K(R) be a maximal compact subgroup of G(R), with complexification K. The algorithm permits the following:

  1. Description of G and G(R)
  2. Geometry of the flag variety K\G/B
  3. Calculation of Cartan subgroups and Weyl groups
  4. Parametrization of irreducible representations
  5. Calculation of Cayley transforms and the cross action
  6. Calculation of Kazhdan-Lusztig polynomials
I will describe at least (1-4) in some detail, and make some remarks about (5-6), time permitting. Some notes will be available at the beginning of the lectures.

These work is part of the Atlas of Lie Groups and Representations, see the Atlas web site. One goal of the project is to compute the unitary dual of G(R). Another goal is to make this information and software available to the general mathematical audience. In particular an early version of the software may be downloaded at the Atlas software page.

For some background reading for these talks see the papers section of the atlas web site. In particular see
Combinatorics for the representation theory of real reductive groups
Parameters for Real Groups
Algorithms for Structure Theory

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