Talk by Robert Kurinczuk, University of Sheffield, UK.
Representations of a group describe its possible actions by linear transformations; and classifying the representations of a given group is a central question in algebra. For matrix groups over finite fields, one approach to understanding these representations is to build them up from representations of smaller matrix groups, a process studied by Harish-Chandra. For example, representations of general linear groups can be built up from representations of products of smaller general linear groups. The irreducible representations which cannot be obtained in this way are called supercuspidal, and Harish-Chandra's theory shows how to build up all representations from these basic building blocks.
A natural question is whether every irreducible representation determines a unique collection of these building blocks. For general linear groups, the answer is yes: this was shown by Harish-Chandra for complex representations, and Vignerás proved it still holds even when the representations have coefficients in a field of positive characteristic, where many familiar tools of representation theory no longer apply. However, for other matrix groups (such as symplectic groups), uniqueness can fail.
We will introduce this construction and discuss what governs uniqueness, presenting a new class of examples where it holds. We will also explain the analogous question for matrix groups over the p-adic numbers and, time permitting, its connection with the local Langlands correspondence.
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