# Representation theory seminar 2023 Semester 1

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

**Place and time: **Peter Hall 162, Thursdays 2:15 – 4:15pm

June 29: **Iva Halacheva** (Northeastern University) Spectrum of type A Bethe algebras and Gelfand-Tsetlin patterns

*Place and time*: Peter Hall 162, 2:15pm – 3:15pm

Bethe algebras are a family of maximal commutative subalgebras of the Yangian, parametrized by regular elements of the maximal torus. For the Yangian of gl(n), this family is known to extend to the Deligne-Mumford compactification of M(0,n+2), and after taking real points the corresponding Bethe subalgebras act with simple spectrum on a fixed tame representation of the Yangian. This leads to a covering, depending on the choice of the representation, with fiber over a point x in the compactification given by the spectrum of the Bethe algebra B(x). I will describe the spectrum in terms of Gelfand-Tsetlin keystone patterns (or alternatively skew semistandard Young tableaux) and discuss the resulting monodromy action. This is joint work with Anfisa Gurenkova and Lenya Rybnikov.

May 18: **Arun Ram** (University of Melbourne) Lusztig varieties and Macdonald polynomials

*Place and time*: Peter Hall 162, 2:15pm – 3:15pm

In recent works Abreu-Nigro and Xuhua He have introduced the term Lusztig variety. I like this term, as Lusztig has many papers about these varieties. In 1997 Halverson and I computed the number of points of Type A nilpotent Lusztig varieties over finite fields in connection to characters of Hecke algebras. Recently, my study of Macdonald polynomials and central elements in Hecke algebras have led me to look at these computations again.

May 11: **Naoki Genra** (University of Tokyo) Reduction by stages on W-algebras

*Place and time*: Peter Hall 162, 2:15pm – 3:15pm

Let be a Poisson variety with a Hamiltonian -action and be a normal subgroup of . Then is obtained by a (Hamiltonian) reduction of by the induced -action under suitable assumptions, called reduction by stages. We apply for the Slodowy slices and show that the Slodowy slice associated to is obtained by a reduction of the Slodowy slice associated to for a simple Lie algebra and nilpotent orbits , such that with some conditions. The quantum cases imply that the finite/affine W-algebras associated to are obtained by W-algebras associated to , which proves a conjecture of Morgan in finite cases and gives a conjectural generalization of results of Madsen and Ragoucy in affine cases. This is a joint work with Thibault Juillard.

May 4: **Peter Fiebig **(Erlangen) Representations and Binomial Coefficients

*Place and time*: Peter Hall 162, 2:15pm –

The talk is motivated by the „generational phenomena” that occur in the representation theory of algebraic groups in positive characteristics. The representation theory of quantum groups is known to provide a first step approximation to modular representations. Lusztig was the first to suggest that there should be „algebraic structures” that provide further steps towards modular representations beyond quantum groups. None of these structures are known today, even though some candidates have been suggested by several authors. In the talk I want to motivate the generational idea and then introduce a model category that makes the proximity of modular and quantum representations quite transparent. Using this category I want to show that the generational problem seems to be closely connected to finding generalizations of binomial coefficients.

April 27: **Alexandre Minets** (University of Edinburgh) A proof of P=W conjecture

*Place and time*: Peter Hall 162, 2:15pm – 4:15pm

Let C be a smooth projective curve. The non-abelian Hodge theory of Simpson is a homeomorphism between the character variety M_B of C and the moduli of (semi)stable Higgs bundles M_D on C. Since this homeomorphism is not algebraic, it induces an isomorphism of cohomology rings, but does not preserve finer information, such as the weight filtration. Based on computations in small rank, de Cataldo-Hausel-Migliorini conjectured that the weight filtration on H^*(M_B) gets sent to the perverse filtration on H^*(M_D), associated to the Hitchin map. In this talk, I will explain a recent proof of this conjecture, which crucially uses the action of Hecke correspondences on H^*(M_D). Based on joint work with T. Hausel, A. Mellit, O. Schiffmann.

Monday April 17: **Xuhua He** (Chinese University of Hong Kong) Affine Deligne-Lusztig varieties and affine Lusztig varieties

*Place and time*: Peter Hall 162, 4pm – 5pm

*Abstract*: Roughly speaking, an affine Deligne-Lusztig variety describes the intersection of a given Iwahori double coset with a Frobenius-twisted conjugacy class in the loop group; while an affine Lusztig variety describes the intersection of a given Iwahori double coset with an ordinary conjugacy class in the loop group. The affine Deligne-Lusztig varieties provide a group-theoretic model for the reduction of Shimura varieties and play an important role in the arithmetic geometry and Langlands program. The affine Lusztig varieties encode the information of the orbital integrals of Iwahori-Hecke functions and serve as building blocks for the (conjectural) theory of affine character sheaves. In this talk, I will explain a close relationship between affine Lusztig varieties and affine Deligne-Lusztig varieties, and consequently, provide an explicit nonemptiness pattern and dimension formula for affine Lusztig varieties in most cases. This talk is based on my preprint arXiv:2302.03203.

April 13: **Dennis Gaitsgory** (Max Planck Institute for Mathematics) Categorical geometric Langlands for D-modules

*Place and time*: Peter Hall 162, 2:15pm – 4:15pm

*Abstract*: In the talk we will describe the recently obtained proof of GLC in the context of D-modules. This is a joint project with D.Arikinkn, D. Beraldo, L.Chen, J.Faegerman, K. Lin, and S.Raskin, who made the most crucial contributions.

April 12: **Dennis Gaitsgory** (Max Planck Institute for Mathematics) Geometric Langlands with nilpotent singular support

*Place and time*: Peter Hall 162, 2:15pm – 4:15pm

*Abstract*: This talk will summarize the recent series of papers by Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky.

We’ll introduce a version of the categorical geometric Langlands conjecture that makes sense for l-adic sheaves over a base of positive characteristic. We will explain the mechanism of the categorical trace of Frobenius, which allows to pass from the geometric assertion to a classical one.

March 23: **Changlong Zhong** (State University of New York at Albany) Oriented cohomology of the affine Grassmannian and the Peterson subalgebra

*Place and time*: Peter Hall 162, 2:15pm – 3:15pm

*Abstract: *Peterson subalgebra is a subalgebra of the (small) torus-equivariant homology of the affine Grassmannian. It was proved by Peterson and Lam-Shimozono that certain localization of this algebra is isomorphic to the quantum cohomology of the flag variety. K-theoretic analogue of this result was recently proved by Kato, as part of his study of semi-infinite flag varieties. In this talk I will talk about generalization of the Peterson subalgebra to the oriented cohomology in the sense of Levine-Morel. I will then talk about a certain localization of this subalgebra in the case of connective K-theory.

March 16: **Aritra Bhattacharya** (Institute of Mathematical Sciences, Chennai) Haglund’s positivity conjecture for Macdonald polynomials

*Place and time*: Peter Hall 162, 2:15pm – 3:15pm

*Abstract*: The Macdonald symmetric functions are an incredible family of symmetric functions that simultaneously generalize many known bases of symmetric functions, such as the Schur functions and the Hall-Littlewood functions. The transition matrix between the Hall-Littlewood and the Schur functions are very well studied – they are given by the Kostka-Foulkes polynomials which are polynomials in with non-negative integer coefficients. However very little is known about the transition matrix between the Macdonald functions and the Schur functions.

Haglund conjectured that the Schur coefficients of the integral form Macdonald Polynomials have the positivity property that . We present some new partial results about this conjecture.

March 9: **Alistair Savage** (University of Ottawa) Diagratification

*Place and time*: Peter Hall 162, 2:15pm – 3:15pm

*Abstract*: We will explain how one can construct diagrammatic presentations of categories of representations of Lie groups and their associated quantum groups using only a small amount of information about these categories. To illustrate the technique in concrete terms, we will focus on the exceptional Lie group of type F4.

March 2: **Kari Vilonen** (University of Melbourne) Introductory remarks on geometric Langlands

*Place and time*: Peter Hall 162, 2:15pm – 4:15pm

*Abstract*: In preparation for the talks of Gaitsgory in April I will explain some aspects of the geometric Langlands program.

Some related notes by Ed Frenkel (sent in by Lance Gurney)

January 19: **Ben Webster** (University of Waterloo) Noncommutative resolutions and Coulomb branches

*Place and time*: Peter Hall 107, 2pm – 4pm

*Abstract*: Work of Braverman, Finkelberg and Nakajima provides a new window onto the world of noncommutative algebras, by constructing both new and previously known algebras as Coulomb branches. While beautiful, their construction involves some very infinite dimensional geometry. I’ll explain how we can replace this with a combinatorial construction, and how this leads us to a new perspective on noncommutative resolutions and categories of coherent sheaves.

January 12: **Emanuel Scheidegger** (Peking University) Aspects of modularity for Calabi-Yau threefolds

*Place and time*: Peter Hall 107, 2pm – 3pm

*Abstract*: We give an overview of some mostly conjectural aspects of modularity for Calabi-Yau threefolds. We focus on one parameter families of hypergeometric type and give computational results in terms of classical modular forms. If time permits, we show an explicit correspondence in one case. This is based on work with K. Boenisch, A. Klemm, and D. Zagier, 2203.09426.

January 5: **Lisa Carbone **(Rutgers University) A Lie group analog for the monster Lie algebra

*Place and time*: Peter Hall 107, 2pm – 3pm

*Abstract*: Let be the monster Lie algebra, a generalized Kac–Moody algebra. We describe various approaches to constructing an analog of a Lie group associated to . We construct a group , associated to , given by generators and relations. We also construct a group of automorphisms of a completion of . The subgroup of generated by all positive imaginary roots embeds in .