The theory of types in reductive p-adic groups and the related Hecke algebras

by Shaun Stevens, University of East Anglia

The category of smooth complex representations of a connected reductive group over a p-adic field can be decomposed as a direct product of subcategories using parabolic induction. The theory of types (Bushnell-Kutzko) aims to describe these subcategories as module categories over a Hecke algebra, generalizing the work of Borel for representations with an Iwahori-fixed vector. The existence of types is not known in general but there are now many examples.

I will review: the theory of types and covers; the Hecke algebras involved; what phenomena can occur; and what this can tell us about parabolic induction.