Hecke algebras play a fundamental role in the representation theory of $p$-adic groups. Historically, the first Hecke algebra to be well understood was the Iwahori–Hecke algebra, the one associated to the “trivial” Bernstein block containing the trivial representation. In a similar vein, Hecke categories of loop groups are important in many questions in modern (geometric) representation theory. So far, only the Iwahori–Hecke category of a loop group has been extensively studied. The goal of this course, however, is to say something about the Hecke algebras and categories beyond the Iwahori level. We will briefly go over the classical theory about Hecke algebras, and then concentrate on some more recent theories about Hecke categories.
Our main tool to study these Hecke categories will be a version of Soergel’s theory of bimodules. It turns out to construct them, the more natural framework is to work in the Kac–Moody setting, and to include twistings from the quantum/metaplectic Langlands.
Some possible topics include:
- Depth zero Hecke algebras
- (Monodromic) affine Hecke categories
- Kloosterman sheaves
- Soergel bimodule
- Applications to quantum local geometric Langlands
- Applications to tame local geometric Langlands
- Beyond tame level