Tag - Geometric representation theory

Lucas Mason-Brown: What is a Unipotent Representation?

22nd February, 2021

The concept of a unipotent representation has its origins in the representation theory of finite Chevalley groups. Let G(𝔽q) be the group of 𝔽q-rational points of a connected reductive algebraic group G. In 1984, Lusztig completed the classification of irreducible representations of G(𝔽q). He showed:

   1.  All irreducible representations of G(𝔽q) can be constructed from a finite set of building blocks -- called 'unipotent representations.'
   2.  Unipotent representations can be classified by certain geometric parameters related to nilpotent orbits for a complex group associated to G(𝔽q).

Now, replace 𝔽q by ℂ, the field of complex numbers, and replace G(𝔽q) with G(ℂ). There is a striking analogy between the finite-dimensional representation theory of G(𝔽q) and the unitary representation theory of G(ℂ). This analogy suggests that all unitary representations of G(ℂ) can be constructed from a finite set of building blocks - called 'unipotent representations' - and that these building blocks are classified by geometric parameters related to nilpotent orbits. In this talk I will propose a definition of unipotent representations, generalizing the Barbasch-Vogan notion of 'special unipotent'. The definition I propose is geometric and case-free. After giving some examples, I will state a geometric classification of unipotent representations, generalizing the well-known result of Barbasch-Vogan for special unipotents.

Chris Chung: iQuantum Covering Groups: Serre presentation and canonical basis

26th January, 2021

In 2016, Bao and Wang developed a general theory of canonical basis for quantum symmetric pairs (U,Ui), generalizing the canonical basis of Lusztig and Kashiwara for quantum groups and earning them the 2020 Chevalley Prize in Lie Theory. The i-divided powers are polynomials in a single generator that generalize Lusztig's divided powers, which are monomials. They can be similarly perceived as canonical basis in rank one, and have closed form expansion formulas, established by Berman and Wang, that were used by Chen, Lu and Wang to give a Serre presentation for coideal subalgebras Ui, featuring novel i-Serre relations when τ(i)=i. Quantum covering groups, developed by Clark, Hill and Wang, are a generalization that `covers' both the Lusztig quantum group and quantum supergroups of anisotropic type. In this talk, I will talk about how the results for i-divided powers and the Serre presentation can be extended to the quantum covering algebra setting, and subsequently applications to canonical basis for Uiπ, the quantum covering analogue of Ui, and quantum covering groups at roots of 1.

Laura Rider: Modular Perverse Sheaves on the Affine Flag Variety

7th December, 2020

There are two categorical realizations of the affine Hecke algebra: constructible sheaves on the affine flag variety and coherent sheaves on the Langlands dual Steinberg variety. A fundamental problem in geometric representation theory is to relate these two categories by a category equivalence. This was achieved by Bezrukavnikov in characteristic 0 about a decade ago. In this talk, I will discuss a first step toward solving this problem in the modular case joint with R. Bezrukavnikov and S. Riche.

Harrison Chen: Coherent Springer theory and categorical Deligne-Langlands

23rd November, 2020

Kazhdan and Lusztig proved the Deligne-Langlands conjecture, a bijection between irreducible representations of unipotent principal block representations of a p-adic group with certain unipotent Langlands parameters in the Langlands dual group (plus the data of certain representations). We lift this bijection to a statement on the level of categories. Namely, we define a stack of unipotent Langlands parameters and a coherent sheaf on it, which we call the coherent Springer sheaf, which generates a subcategory of the derived category equivalent to modules for the affine Hecke algebra (or specializing at q, unipotent principal block representations of a p-adic group). Our approach involves categorical traces, Hochschild homology, and Bezrukavnikov's Langlands dual realizations of the affine Hecke category.

Reuven Hodges: Coxeter combinatorics and spherical Schubert geometry

16th November, 2020

This talk will introduce spherical elements in a finite Coxeter system. These spherical elements are a generalization of Coxeter elements, that conjecturally, for Weyl groups, index Schubert varieties in the flag variety G/B that are spherical for the action of a Levi subgroup. We will see that this conjecture extends and unifies previous sphericality results for Schubert varieties in G/B due to P. Karuppuchamy, J. Stembridge, P. Magyar–J. Weyman-A. Zelevinsky. In type A, the combinatorics of Demazure modules and their key polynomials, multiplicity freeness, and split-symmetry in algebraic combinatorics are employed to prove this conjecture for several classes of Schubert varieties.

Nicolas Libedinsky: On Kazhdan-Lusztig theory for affine Weyl groups

5th November, 2020

Kazhdan-Lusztig polynomials are a big mystery. On a recent work with Leonardo Patimo (following Geordie Williamson) we were able to calculate them explicitly for affine A2. We dream of a similar description for all affine Weyl groups, but it seems like an incredibly difficult program. I will explain some new results in this direction and what we believe that is doable. Another part of this project is to produce an approach towards the following question: for a given element in an affine Weyl group, what are the prime numbers p such that the p-canonical basis is different from the canonical basis?

Nicolle Gonzalez: A skein-theoretic Carlsson-Mellit algebra

2nd November, 2020

The shuffle conjecture was a big open problem in algebraic combinatorics which gave a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. This conjecture was finally solved after 14 years by Carlsson and Mellit by the introduction of a new interesting algebra denoted Aq,t. This algebra arises as an extension of the affine Hecke algebra by certain raising and lowering operators and acts on the space of symmetric functions via certain complicated plethystic operators. In later work by Carlsson, Mellit, and Gorsky this algebra and its representation was realized using parabolic flag Hilbert schemes and was also shown to contain the generators of the elliptic Hall algebra. I will discuss a new topological formulation of Aq,t and its representation over a thickened annulus and a categorification thereof over the derived trace of the Soergel category. This is joint work with Matt Hogancamp.

Daniel Tubbenhauer: On categories of tilting modules

12th October, 2020

In this talk I will report on the progress in the project of trying to understand categories of tilting modules as categories, meaning the morphisms (and not objects) in these categories and their relations, with the focus being on SL2 and SL3. Joint work with Paul Wedrich.

Arun Ram: Integrable modules for affine Lie algebras

5th October, 2020

These modules naturally divide themselves into three categories: positive level, negative level and level 0. The positive level modules are highest weight, the negative level ones are lowest weight, and the level 0 ones are neither. But all three classes of modules have some nice character formulas, a good crystal theory in the sense of Kashiwara-Lusztig-Littelmann, and Borel-Weil-Bott type geometric constructions. The geometric constructions use, respectively, the thin affine flag variety (for positive level), the thick affine flag variety (for negative level), and the semi-infinite flag variety (for level 0).

Eugene Gorsky: Link homology and Hilbert schemes

28th September, 2020

The category of Soergel bimodules categorifies the Hecke algebra. Khovanov and Rozansky used Soergel bimodules to define triply graded link homology (also known as Khovanov-Rozansky homology) which categorifies HOMFLY-PT link invariant. I will survey some results and conjectures relating Soergel bimodules and Khovanov-Rozansky homology with the Hilbert scheme of points on the plane. The talk is based on joint works with Matt Hogancamp, Andrei Negut, Jake Rasmussen and Paul Wedrich.