A recurrent theme in representation theory is the study of representations of entire families of groups, capitalizing on the interactions between them. The first incarnation of this approach dates back to the origins of representation theory, with the study of representations of the symmetric groups via induction and restriction. Harish-Chandra adapted these ideas to other families of groups, such as semisimple Lie groups. His approach, known as the Harish-Chandra philosophy of cusp forms, was subsequently applied to further families, including finite groups of Lie type, among many others.

In this lecture, I will describe a variant of these ideas that is suitable for profinite groups. All necessary ingredients will be defined. This is joint work with Tyrone Crisp and Ehud Meir.