Tag - Geometric representation theory

Daniel Orr: Difference Operators for Wreath Macdonald Polynomials

17th October, 2021

very concrete auxiliary algebraic structures that were constructed in order to define them. Later, when Haiman's proof of the Macdonald positivity conjecture revolutionized the subject, the scope of Macdonald theory widened to include the geometry of Hilbert schemes of points in the plane. (For this reason, one should associate ordinary Macdonald polynomials with the Jordan quiver.)

A cyclic quiver generalization of Macdonald polynomials was born in reverse, starting with a geometric conjecture which was made by Haiman and later proved by Bezrukavnikov and Finkelberg. Thus the resulting polynomials, which are known as wreath Macdonald polynomials, arise from the geometry of cyclic quiver Nakajima varieties. Their existence relies on an elusive object known as the Procesi bundle, which is available only by deep and indirect means.

Only recently has direct understanding of wreath Macdonald polynomials begun to emerge, through methods based on the quantum toroidal algebra. In this talk, I will review the origins of (wreath) Macdonald theory and discuss new explicit results on wreath Macdonald polynomials, and anticipated applications, from joint work in progress with Mark Shimozono and Joshua Wen.

William Graham: Tangent spaces and T-invariant curves of Schubert varieties

20th September, 2021

The set of weights of T-invariant curves to a Schubert variety at a T-fixed point admits a simple description due to Carrell and Peterson. The set of weights to the tangent space is much more complicated to describe, and has only been explicitly described in classical types. The main result is that although these two sets of weights are different, they generate the same cone in the dual of the Lie algebra of T.

Tamanna Chatterjee: Parity Sheaves Arising from Graded Lie Algebras II

13th September, 2021

Let G be a complex, connected, reductive, algebraic group, and Ο‡ : β„‚Γ— β†’ G be a fixed cocharacter that defines a grading on 𝔀, the Lie algebra of G. In my first talk I have talked about the grading, derived category of equivariant perverse sheaves, bijection between the simple objects and some pairs that we are familiar with. In positive characteristic parity sheaves will play an important role. In this talk I will define parabolic induction and restriction both on nilpotent cone and graded setting. We will dive into the results of Lusztig in characteristic 0 in the graded setting. Under some assumptions on the field k and the group G we will recover some results of Lusztig in characteristic 0. These assumptions together with Mautner's cleanness conjecture will play a vital role. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspidal pair. Lusztig's work on β„€-graded Lie algebras is related to representations of affine Hecke algebras, so a long term goal of this work will be to interpret parity sheaves in the context of affine Hecke algebras.

Tamanna Chatterjee: Parity Sheaves Arising from Graded Lie Algebras I

30th August, 2021

Let G be a complex, connected, reductive, algebraic group, and Ο‡ : β„‚Γ— β†’ G be a fixed cocharacter that defines a grading on 𝔀, the Lie algebra of G. Let G0 be the centralizer of Ο‡(β„‚Γ—). Here I will talk about G0-equivariant parity sheaves on the n-graded piece, 𝔀n. For the first half we will spend on building the background of derived category of equivariant perverse sheaves, bijection between the simple objects and some pairs that we are familiar with. In positive characteristic parity sheaves will play an important role. We want to study DbG0(𝔀n, k) for characteristic of k is positive. For that we will dive into the results of Lusztig in characteristic 0 in the graded setting. The main result from Lusztig is that every perverse sheaf occurs as a direct summand of the parabolic induction of the simple perverse sheaf associated to some cuspidal pair. The goal of the second talk will be to extend this result into positive characteristic.

Geordie Williamson: Spectra in representation theory

18th June, 2021

In geometric representation theory cohomology, intersection cohomology and constructible sheaves show up everywhere. This might seem strange to an algebraic topologist, who might ask: why this emphasis on cohomology, when there are so many other interesting cohomology theories (like K-theory, elliptic cohomology, complex cobordism, ...) out there? They might also ask: is there something like "intersection K-theory", or "intersection complex cobordism"? This is something I've often wondered about. I will describe work in progress with Ben Elias, where we use Soergel bimodules to investigate what KU-modules look like on the affine Grassmannian. We have checked by hand that in types A1, A2 and B2, one gets something roughly resembling the quantum group. Speaking very roughly, the intersection K-theory of Schubert varieties in the affine Grassmannian should recover the irreducible representations of the quantum group. Inspirations for this work include a strange Cartan matrix discovered by Ben Elias, and work of Cautis-Kamnitzer.

Max Gurevich: New constructions for irreducible representations in monoidal categories of type A

28th May, 2021

One ever-recurring goal of Lie theory is the quest for effective and elegant descriptions of collections of simple objects in categories of interest. A cornerstone feat achieved by Zelevinsky in that regard, was the combinatorial explication of the Langlands classification for smooth irreducible representations of p-adic GLn. It was a forerunner for an exploration of similar classifications for various categories of similar nature, such as modules over affine Hecke algebras or quantum affine algebras, to name a few. A next step - reaching an effective understanding of all reducible finite-length representations remains largely a difficult task throughout these settings.

Recently, joint with Erez Lapid, we have revisited the original Zelevinsky setting by suggesting a refined construction of all irreducible representations, with the hope of shedding light on standing decomposition problems. This construction applies the Robinson-Schensted-Knuth transform, while categorifying the determinantal Doubilet-Rota-Stein basis for matrix polynomial rings appearing in invariant theory. In this talk, I would like to introduce the new construction into the setting of modules over quiver Hecke (KLR) algebras. In type A, this category may be viewed as a quantization/gradation of the category of representations of p-adic groups. I will explain how adopting that point of view and exploiting recent developments in the subject (such as the normal sequence notion of Kashiwara-Kim) brings some conjectural properties of the RSK construction (back in the p-adic setting) into resolution. Time permits, I will discuss the relevance of the RSK construction to the representation theory of cyclotomic Hecke algebras.

Anna Romanov: A categorification of the Lusztig-Vogan module

26th April, 2021

Admissible representations of real reductive Lie groups are a key player in the world of unitary representation theory. The characters of irreducible admissible representations were described by Lusztig-Vogan in the 80s in terms of a geometrically defined module over the associated Hecke algebra. In this talk, I'll describe a categorification of this module using Soergel bimodules, with a focus on examples.

Evgeny Mukhin: Supersymmetric analogues of partitions and plane partitions

8th April, 2021

We will explain combinatorics of various partitions arising in the representation theory of quantum toroidal algebras associated to Lie superalgebra 𝔀𝔩(m|n). Apart from being interesting in its own right, this combinatorics is expected to be related to crystal bases, fixed points of the moduli spaces of BPS states, equivariant K-theory of moduli spaces of maps, and other things.

William Graham: Tangent spaces and T-stable curves in Schubert varieties

22nd March, 2021

The set of T-stable curves in a Schubert variety through a T-fixed point is relatively easy to understand, but the tangent space is more difficult. In this talk we describe some new relations between the tangent space and the set of T-fixed curves. This is joint work with Victor Kreiman.

Catharina Stroppel: Verlinde rings and DAHA actions

16th March, 2021

In this talk we will briefly recall how quantum groups at roots give rise Verlinde algebras which can be realised as Grothendieck rings of certain monoidal categories. The ring structure is quite interesting and was very much studied in type A. I will try to explain how one gets a natural action of certain double affine Hecke algebras and show how known properties of these rings can be deduced from this action and in which sense modularity of the tensor category is encoded.