# Seminar

## Representation theory seminar 2022 Semester 2

Organisers: Kari Vilonen, Ting Xue

Topics: TBA.

Time: Thursdays 3:15pm-5pm.

Location: Peter Hall 107

### Schedule

Aug 18 Volker Heiermann (Université d’Aix-Marseille) Affine Hecke Algebras, Intertwining Operators and Types

To put into relation a Bernstein component of the category of complex smooth representations of a p-adic reductive group and representations of Affine Hecke Algebras, one can in principle use two methods: the theory of types of Bushnell-Kutzko or the intertwining algebra of a certain projective generator of this category. From the first method, it is rather clear that unitarity is preserved, which is less immediate for the second method. The aim of the talk is to explain the situation, as time permits.

Aug 11 Matthew Emerton (Chicago) Categorical perspectives on the arithmetic Langlands program

A categorical perspective, which has long been the norm in the geometric Langlands program, has recently emerged in the arithmetic Langlands program as well.   I will explain some of the ideas related to this perspective, including the Fargues–Scholze conjecture; conjectures and results (due variously to Ben-Zvi–Chen–Helm–Nadler, Hellmann, Zhu, as well as the speaker and Toby Gee) regarding fully faithful functors from categories of representations of p-adic reductive groups to categories of coherent sheaves on stacks of Langlands parameters; and conjectural descriptions of the cohomology of locally  symmetric congruence quotients (e.g. Shimura varieties) in terms of these categorical ideas.

Aug 4 joint Representation Theory/Number theory seminar

Chen Wan (Rutgers) A multiplicity formula of K-types

In this talk, by using the trace formula method, I will prove a multiplicity formula of K-types for all representations of real reductive groups in terms of the Harish-Chandra character.

Jul 28 David Ridout (Melbourne) Inverse quantum hamiltonian reduction

Let g be a (complex, finite-dimensional) simple Lie algebra.  Quantum hamiltonian reduction is a cohomological functor from category O for an affine vertex algebra $V_k(g)$ to the corresponding category for a W-algebra $W_k(g;f)$.  This functor depends on the orbit of a nilpotent f in g, as does the W-algebra.  Unfortunately, it is only well understood for regular and minimal orbits.

A new approach to W-algebra module categories starts with Adamovic’s recent observation that there exist functors going the other way: from the W-algebra to the affine vertex algebra (or more typically, to another W-algebra).  Interestingly, these functors do not naturally land in category O but in a much larger category.  More interestingly, it appears that this approach will work for general nilpotent orbits.

We currently only understand these “inverse functors” in a few examples.  I will attempt to explain some of what we know (and maybe some of what we believe) when $g=sl_2$.

2022 Semester 1

2021 Semester 2

2021 Semester 1

2020 Semester 2