SMRI Seminar: Chan -- Perverse sheaves and representations of p-adic groups

Speaker: Charlotte Chan, University of Michigan 

Abstract: One of the first basic ideas we all learn is that a continuous function is
determined by its values on a dense open subset.  In representation theory, this allows
us to recognize a representation of a Lie group from an especially well-behaved
locus—that o f regular semisimple elements.  But what if we want to study
representations of matrix groups over finite fields? Lusztig’s revolutionary idea in
the 1980s was that intermediate extension—the algebro-geometric version of the
analytic notion of limit—applies in representation theory in discrete settings.  I
will explain this picture and describe a recent construction (joint with R.
Bezrukavnikov) of perverse sheaves that give rise to positive-depth supercuspidal
representations of p-adic groups.  In the simplest nontrivial case, this resolves a
conjecture of Lusztig.