Seminar
Representation theory seminar 2025 Semester 2
Organisers: Dougal Davis, Kari Vilonen, Ting Xue
Place and time: Peter Hall 162, Tuesdays, 1:30-3PM
Please contact one of the organisers to be added to the mailing list.
Upcoming Talks
12 Aug Ivan Losev (Yale) Harish-Chandra center for affine Kac-Moody algebras in positive characteristic
Abstract: This talk is based on a joint work in preparation with Gurbir Dhillon. A remarkable theorem of Feigin and E. Frenkel from the early 90’s describes the center of the universal enveloping algebra of an (untwisted) affine Kac-Moody Lie algebra at the so called critical level proving a conjecture of Drinfeld: the center in question is the algebra of polynomial functions on an infinite dimensional affine space known as the space of opers. In our work we study a part of the center in positive characteristic p at an arbitrary non-critical level. Namely, we prove that the loop group invariants in the completed universal enveloping algebra is still the algebra of polynomials on an infinite dimensional affine space that is “p times smaller than the Feigin-Frenkel center”. In my talk I plan to introduce all necessary notions, state the result, explain motivations and examples.
13 Aug 2:30-4pm Ivan Losev (Yale) Categorical Heisenberg actions and modular representations of rational Cherednik algebras
Abstract: This is based on arXiv:2408.02485, joint with Bezrukavnikov. We construct certain functors on categories of representations of rational Cherednik algebras associated with symmetric groups in zero and large positive characteristic. Our functors are indexed by pairs of a partition and a rational number, the slope. For a given slope, the functors give an action of the positive half of the Heisenberg Lie algebra, while when the slope varies and the characteristic is positive we get a categorical action of the positive half of the elliptic Hall algebra. In this talk I will define rational Cherednik algebras and their categories of modular representations, state our main result and explain how our construction affinizes a version of the Steinberg tensor product theorem for the quantum GL_n. Time permitting, I’ll say a couple of words about the construction, a categorical representation of the positive half of the elliptic Hall algebra, etc.
19 Aug Edmund Heng (Sydney) Fusion quivers and Coxeter groups
Abstract: The study of module categories over fusion categories have focussed mostly on the semisimple ones. In this talk I will introduce the notion of fusion quivers and their representations, the categories of which form hereditary (global projective dimension 1) abelian module categories over fusion categories. This naive “one-step”
generalisation from semisimple module categories uncovers a wealth of interesting new connections to Coxeter theory. In particular, I will present a classification result in the spirit of Gabriel: the finite-representation-type fusion quivers are now classified by the Coxeter—Dynkin diagrams: these include the (crystallographic) Dynkin diagram from Lie algebras, and perhaps surprisingly, also the non-crystallographic diagrams H and I, which all together classify the finite Coxeter groups. If time allows, I will also discuss some applications to Coxeter and Artin—Tits groups.
5 Aug Matt Emerton (Chicago) A local-global compatibility for cohomology of Hilbert modular varieties
Abstract: The speaker, together with Xinwen Zhu, has proposed a conjecture that describes the cohomology of Shimura varieties with coefficients in an automorphic local system in terms of the conjectural categorical local Langlands correspondence (so this is a conjecture which depends on a conjecture!). Eugen Hellmann has proposed a formula for the values of the p-adic categorical local Langlands correspondence in the case of locally algebraic types, after inverting p. In this talk we will state, and sketch the proof of, a result in the direction of our conjecture with Zhu in the Hilbert modular case, using Hellmann’s proposal for the relevant cases of the p-adic categorical local Langlands correspondence. The main inputs in the proof are the de Rham comparison theorem between p-adic etale and p-adic de Rham cohomology, along with the elementary theory of q-expansions of Hilbert modular forms.