# Representation theory seminar 2022 Semester 1

## Representation theory seminar 2022 Semester 1

**Organisers**: Kari Vilonen, Ting Xue, Yaping Yang

**Topics: **This semester our learning seminar will focus on real groups. On alternate weeks we will have research talks by the members of our group.

**Time:** Thursdays 3:15pm-5pm.

Location: Peter Hall 101 and Zoom

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

### Schedule

Jun 9 **Arun Ram **Is there a Kac-Moody-like presentation of toroidal algebras?

(Evan Williams Theatre)

Abstract: Ion-Sahi have pointed to a Coxeter like presentation of the double affine Artin group (DAArt). I will explain how this presentation could be discovered from a matrix representation of the double affine Weyl group (DAWG) which naturally exhibits the action of (acting on the DAWG) by automorphisms. The position of the Heisenberg group inside the DAWG is clearly visible in this representation. The Coxeter-like presentation uses three affine Dynkin diagrams of the same type glued together along the common finite Dynkin diagram and a single additional “superglue” relation. I wonder if these results could be extended to provide a Kac-Moody-like presentation of quantum toroidal algebras.

Jun 2 **Peter Hall 213**

**3-4pm** **Wille Liu** (Max-Planck-Institut für Mathematik) Trigonometric Knizhnik-Zamolodchikov functor and affine Hecke algebra

Abstract: The Knizhnik–Zamolodchikov (KZ) equations arised from the theory of Wess-Zumino-Witten conformal blocks on the Riemann sphere associated with a Lie group. Inspired by the Schur–Weyl duality, Cherednik introduced in the 90s a version of KZ equations for graded affine Hecke algebras. The operators (called Dunkl operators) that show up in the KZ equations can be conveniently organised into an associative algebra, called trigonometric double affine Hecke algebra (DAHA).

In this talk, I will explain certain representation-theoretic aspect of the trigonometric DAHA with emphasis on its KZ equations.

**4-5pm Cheng-Chiang Tsai **(Academia Sinica) Harmonic analysis of p-adic groups and affine Springer theory

Abstract: In this presentation we give a survey about less well-known connections between harmonic analysis of p-adic groups and affine Springer theory. One upshot is the so-called “homogeneity property” in harmonic analysis which describes the asymptotic behavior of the (co)homology affine Springer fiber as the dimension increases. The other upshot is the theory of endoscopy, which features global methods in number theory, has been developed a lot on the side of p-adic groups since the 80s, and occasionally serves as heuristic for graded and affine Springer theory.

May 26 **Travis Scrimshaw** (Osaka City University) Probability measures from representation theory

Howe duality for the general linear group can be described as the fact that the biregular representation of , where it acts on itself on the right and left, decomposes into a multiplicity free sum of representations. In terms of characters, this yields the Cauchy identity. By dividing both sides by the product, we obtain the famous Schur (probability) measure on partitions. In this talk, we will examine a similar measure constructed from skew Howe duality and discuss the relationship with Krawtchouk polynomials, a family of classical orthogonal polynomials in the Askey scheme. No prior background knowledge will be assumed. This is based on joint work with Anton Nazarov and Olga Postnova.

May 19 **Gufang Zhao **Shifted symplectic structures, mapping stacks, and virtual fundamental classes

The first half of the talk is an informal review of shifted symplectic stacks following Pantev, Toën, Vaquié, and Vezzosi. Familiar examples in representation theory will be discussed. The second half follows the work of Oh and Thomas to introduce virtual fundamental classes for -1 and -2-shifted symplectic stacks. I will also discuss my work in progress joint with Yalong Cao, where we apply this construction to study quantum cohomology of some moduli spaces.

May 12 **Qixian Zhao **(Utah) Reducibility of standard representations: Examples. Notes

I will rephrase the irreducibility criterion in algebraic terms (i.e. without exponentials), demonstrate the proof of the irreducibility criterion explicitly on concrete examples (SU(2,1), SL(3,R) and SL(2,R)), and answer some questions raised in previous talks along the way.

May 5 **Chenyan Wu **Theta correspondence and poles of Eisenstein series

Let be a cuspidal automorphic representation of a classical group. First I will define two invariants attached to , namely, the lowest occurrence of in the theta correspondence and the location of the maximal pole of an Eisenstein series built from and a character. Then I will show a relation between the two invariants and talk about an implication of this result on certain global Arthur packets.

Apr 28 **Qixian Zhao **(Utah) Reducibility of standard representations, continued Notes

Apr 14 **Peter McNamara **Sheaves behaving badly

Given a complex algebraic variety, we construct a canonical sheaf with mod p coefficients that tells us something about the geometry of the singularities and what possible resolutions they have. We discuss how badly these sheaves behave for Schubert varieties. This is motivated

by the desire to study parity sheaves in geometric representation theory.

Apr 7 **Qixian Zhao **(Utah) Reducibility of standard representations Notes

I will present the irreducibility criterion for standard K-equivariant D-modules in the linear case, mostly based on the argument in Hecht-Milicic-Schmid-Wolf. I will present the statement and an outline/idea of proof, and (if time permits) compute the SU(2,1) example.

Apr 7 **Kari Vilonen **Real groups

I will finish the geometrization of Langlands parameters and will introduce the Arthur parameters. I will also make some general remarks about representations leading to the talk of Qixian.

Mar 31 **Yaping Yang ** Quantum groups at roots of 1

I will start with Lusztig’s quantum groups at roots of unity and explain the quantum Frobenius homomorphism and the Steinberg tensor product theorem. I will then talk about a family of quantum groups associate to Morava E-theories. I will also explain the quantum Frobenius homomorphisms among these quantum groups constructed by Gufang and myself. The main ingredient in constructing these Frobenii is the transchromatic character map of Hopkins, Kuhn, Ravenal, and Stapleton. This is based on my joint work with Gufang Zhao.

Mar 24 **Kari Vilonen **Real groups Notes

I will briefly discuss the Fourier transform and nearby cycles in response to questions asked during Ting’s talk on March 17. The main point of this lecture is to explain how to view Langlands parameters for real groups geometrically. This was first explained in a book by Adams, Barbasch, Vogan and in a slightly different formulation in a paper by Adams and du Cloux.

Mar 17 **Ting Xue **Character sheaves and Hecke algebras

We discuss character sheaves in the setting of graded Lie algebras. Via a nearby cycle construction irreducible representations of Hecke algebras of complex reflection groups at roots of unity enter the description of character sheaves. Recent work of Lusztig and Yun relates character sheaves to irreducible representations of trigonometric double affine Hecke algebras. We will explain the connection between the work of Lusztig-Yun and our work, and discuss some conjectures arising from this connection. If time permits, we will discuss applications to cohomology of Hessenberg varieties and affine Springer fibres. This is based on joint work with Kari Vilonen and partly with Tsao-Hsien Chen and Misha Grinberg.

Mar 10 **Kari Vilonen **Real groups Notes

This is the first talk in learning seminar on real groups. In this talk I will give a broad outline of the state of representation theory of real groups. I will also discuss possible future research directions.

Mar 3 Organisational meeting