# Seminar

## Representation theory seminar 2024 Semester 2

**Organisers**: Dougal Davis, Kari Vilonen, Ting Xue

**Place and time: Peter Hall 162, 2:15-4:15PM**

Please contact one of the organisers to be added to the mailing list.

### Upcoming Talks

Aug 29: **Wille Liu** (Academia Sinica) Deligne–Langlands correspondence for affine Hecke algebras at roots of unity

The affine Hecke algebra attached to a root system depends on a complex parameter q. When q is not a root of unity, Kazhdan and Lusztig established in 1987 a geometric parametrisation of simple modules of the affine Hecke algebra (called Deligne–Langlands correspondence). A modified version of this correspondence for root-of-unity q was announced without proof by Grojnowski in 1994. I will explain a proof of it that I discovered, which is based on the study of character sheaves on a graded Lie algebra.

### Past Talks

Aug 8: **Emile Okada** (National University of Singapore) Character sheaves, Hecke algebras, and the wavefront set

The wavefront set is a distribution theoretic invariant first introduced to the study of representations of linear algebraic groups by Roger Howe in the early 80s. It associates to a representation a closed subset of the characteristic variety, and for groups over a local field it contains important arithmetic information. In this talk I will discuss recent progress toward understanding the wavefront set for groups over p-adic fields – the setting which is least well understood. In particular I will present recent developments in harmonic analysis which reduce the study to an arithmetic component, governed in depth 0 by character sheaves, and an algebraic components, governed in depth 0 by branching problems for modules of Hecke algebras.

Aug 6: **Elijah Bodish **(Massachusetts Institute of Technology) Spin link homology

Reshetikhin-Turaev define a Laurent polynomial invariant of knots for each simple Lie algebra “colored” by a finite dimensional irreducible representation. In the case of sl(2) and the defining representation, this polynomial invariant is the Jones polynomial.

Khovanov discovered that the Jones polynomial is the Euler characteristic of a complex of graded vector spaces. Thus, Khovanov’s homology categories the Jones polynomial. Many other definitions of categorified Reshetikhin-Turaev invariants have appeared since. The most notable works are: Khovanov-Rozansky’s generalization of Khovanov homology to sl(n), and Webster’s uniform construction for an arbitrary simple Lie algebra. However, very little is known (e.g. no examples are computed) unless the Lie algebra is sl(n) and the representation is a fundamental representation.

In my talk I will describe how to equip the sl(2n) link homology, colored by the n-th fundamental representation, with an involution such that the (super) Euler characteristic is the so(2n+1) Reshetikhin-Turaev link polynomial. This construction, which is apriori unrelated to Webster’s, is inspired by folding, categorified skew Howe duality, diagrammatics for centralizer algebras, and iquantum groups.

This is based on arXiv:2407.00189 — joint work with Ben Elias and David Rose.

July 23:** Matt Emerton **(U Chicago) Modular forms and Galois representations.

**1-2PM Peter Hall 162**

This is a pre-talk for the Thursday talk aimed at graduate students.

I will begin by recalling the basic facts about Tate modules of elliptic curves. Building on this discussion, I will recall the basic facts about Galois representations attached to modular forms, and sketch the proofs of some of them.

July 25: **Matt Emerton** (U Chicago) From modular curves to categorification

The categorical form of the local Langlands correspondence conjectures (and in some cases proves) that derived categories of smooth representations of p-adic reductive groups can be realized inside the derived categories of coherent sheaves on suitable moduli spaces of Langlands parameters. The goal of this lecture is to motivate this categorical local Langlands conjecture from an arithmetic point of view, with the cohomology of modular curves as the starting point.

June 12: **Shilin Yu **(Xiamen University) Duality of nilpotent orbit covers

3:15 – 4:15pm, Evan Williams Theatre

In their study of special unipotent representations for complex semisimple groups, Barbasch and Vogan constructed a duality map between the nilpotent orbits of and that of its Langlands dual group (also discovered by Lusztig and Spaltenstein), which allows them to describe the special unipotent ideals and representations of in terms of . This duality was later generalized by Sommers, Achar and Losev-Mason-Brown-Matvieievskyi.

In collaboration with Lucas Mason-Brown and Dmytro Matvieievskyi (arXiv:2309.14853), we reinterpret these duality maps in terms of covers of nilpotent orbits. This not only enables the definition of generalized unipotent representations, but also leads to interesting observations and conjectures regarding the birational geometry of the affinizations of the nilpotent orbit covers. If time permits, I will also discuss the connection of these findings to symplectic duality/3d mirror symmetry.