Smith theory and representation theory

Smith theory and representation theory

Geordie Williamson, University of Sydney

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Localizations theorems abound in algebraic topology and algebraic geometry. The idea is that one can sometimes reduce questions about a space X to questions about the fixed points of a group acting on X. The effect is sometimes quite miraculous; replacing, for example, a complicated integral by a simple finite sum. One of the archetypal localization theorems is due to Smith (made somewhere near Princeton in the 30s I believe): he noticed that one has a localization theorem for actions of p-groups on homology spheres. It has since been realized that Smith theory provides a powerful link between cohomology groups with mod p coefficients, and fixed points under p-group actions. Several years ago, David Treumann (a Princeton graduate!) started advocating that Smith theory should have interesting applications in modular representation theory. I will explain an application of these ideas, where we recover fundamental results in the representation theory of algebraic groups via an application of Smith theory to the affine Grassmannian.

This is joint work with Simon Riche.