This talk will be about geometric approaches to the representation theory of reductive algebraic groups in positive characteristic p. A cornerstone of the geometric theory is the geometric Satake equivalence, which gives an incarnation of the category of representations as a category of perverse sheaves on the affine Grassmannian. It is surprising that several "easy" algebraic facts have no explanation on the geometric side. This is frustrating, as several deep conjectures appear to point to a clear relation with the geometry of the affine Grassmannian. One dreams of using geometric Satake as a fundamental localisation theorem (akin to Beilinson-Bernstein localisation for complex semi-simple Lie algebras) from which one can deduce structural results in representation theory. There are now several pieces of evidence in the Langlands program for the viability of this philosophy.

I will explain a recent step towards this dream. One of the "easy" algebraic facts alluded to above is the linkage principle, which decomposes the category of representations into "blocks" controlled by the affine Weyl group. In joint work with Simon Riche, we explain this decomposition via a certain mod p version of hyperbolic localization, known as Treumann-Smith theory. The theory has its roots in Smith's study of Z/pZ actions on spheres in the 1930s, and was upgraded to sheaves a few years ago by Treumann. We also deduce a new proof of the Lusztig character formula (for large p) and a conjecture of ours on characters of tilting modules (for all p).

This is a part of the "GRT at home" seminar series, see grt-home.org

Note the unusual time!