# Past Representation Theory Seminar

We construct triangle equivalences between singularity categories of

two-dimensional cyclic quotient singularities and singularity categories of

a new class of finite dimensional local algebras, which we call Knörrer

invariant algebras. In the hypersurface case, we recover a special case of Knörrer’s equivalence for (stable) categories of matrix factorisations.

We’ll then explain how this led us to study Ringel duality for

certain (ultra strongly) quasi-hereditary algebras.

This is based on joint work with Joe Karmazyn.

The smooth representation theory of a p-adic reductive group G with characteristic zero coefficients is very closely connected to the module theory of its (pro-p) Iwahori-Hecke algebra H(G). In the modular case, where the coefficients have characteristic p, this connection breaks down to a large extent. I will first explain how this connection can be reinstated by passing to a derived setting. It involves a certain differential graded algebra whose zeroth cohomology is H(G). Then I will report on a joint project with R. Ollivier in which we analyze the higher cohomology groups of this dg algebra for the group G = SL_2.

The smooth representation theory of a p-adic reductive group G

with characteristic zero coefficients is very closely connected to the

module theory of its (pro-p) Iwahori-Hecke algebra H(G). In the modular

case, where the coefficients have characteristic p, this connection

breaks down to a large extent. I will first explain how this connection

can be reinstated by passing to a derived setting. It involves a certain

differential graded algebra whose zeroth cohomology is H(G). Then I will

report on a joint project with

R. Ollivier in which we analyze the higher cohomology groups of this dg

algebra for the group G = SL_2.

Enhanced Langlands parameters for a p-adic group G are pairs formed by a Langlands parameter for G and an irreducible character of a certain component group attached to the parameter. We will first introduce a notion

of cuspidality for these pairs. The cuspidal pairs are expected to correspond to the supercuspidal irreducible representations of G via the local Langlands correspondence.

We will next describe a construction of a cuspidal support map for enhanced Langlands parameters, the key tool of which is an extension to disconnected complex Lie groups of the generalized Springer correspondence due to Lusztig.

Finally, we will use this map to decompose the set of enhanced Langlands parameters into Bernstein series.

This is joint work with Ahmed Moussaoui and Maarten Solleveld.