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:

- Description of G and G(R)
- Geometry of the flag variety K\G/B
- Calculation of Cartan subgroups and Weyl groups
- Parametrization of irreducible representations
- Calculation of Cayley transforms and the cross action
- Calculation of Kazhdan-Lusztig polynomials

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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