### Abstract

It's well known that the algebraic geometry of Schubert varieties is closely connected to representation theory for complex reductive Lie algebras; and that many problems in that geometry can be connected to algebra and combinatorics in Weyl groups. Both of these connections persist for representations of real reductive Lie groups. One consequence is that real reductive groups suggest interesting and difficult new problems about Weyl groups. I will sketch very briefly the classical Beilinson-Bernstein localization theory, which makes the connection between representation theory and geometry. I will try to explain more carefully what geometric problems arise, and how they are in turn related to Weyl groups and Hecke algebras. In the process I will introduce the "atlas" computer software, which can work with the combinatorial and algebraic problems. To play with the software in advance, go to: http://atlas.math.umd.edu/software/