Geometric Representation Theory

Harish-Chandra bimodules first appeared in the representation theory of semisimple Lie algebras, and have since then been generalized to quantizations of symplectic singularities, where they categorify (part of) the top Borel-Moore homology of the Steinberg variety. We consider the case of type A rational Cherednik algebras, that in a sense serves as a counterexample for many properties one might expect from the category of HC bimodules. For example, even the number of irreducible objects is highly dependent on the quantization parameter. We will give basic properties of this category, describe its irreducible objects and its relation to the category O and, time permitting, give a complete description via quivers with relations of the block containing the regular bimodule.

Quot schemes of quotients of a trivial bundle of arbitrary rank on a nonsingular projective surface carry perfect obstruction theories and virtual fundamental classes whenever the quotient sheaf has at most 1-dimensional support. The associated generating series of virtual Euler characteristics is conjectured to be a rational function. A similar conjecture can be made about more general descendent series with insertions obtained from the Chern characters of the tautological sheaf. I will discuss several results in this direction obtained jointly with Drew Johnson and Rahul Pandharipande.

Drinfeld proved that the Yangian Yg of a complex semisimple Lie algebra g gives rise to solutions of the quantum Yang-Baxter equations on irreducible, finite-dimensional representations of Yg, which are rational in the spectral parameter. This result was recently extended by Maulik-Okounkov for Kac-Moody algebras corresponding to quivers, and representations arising from geometry.

Surprisingly perhaps, this rationality ceases to hold if one considers arbitrary finite-dimensional representations of Yg, at least if one requires such solutions to be natural with respect to the representation and compatible with tensor products.

I will explain how one can instead produce meromorphic solutions of the QYBE on all finite-dimensional representations by resumming Drinfeld's universal R-matrix R(s) of Yg. The construction hinges on resumming the abelian part of R(s), and on realizing its lower triangular part as a twist conjugating the standard coproduct of Yg to its deformed Drinfeld coproduct.

This is joint work with Sachin Gautam and Curtis Wendlandt, and is based on arXiv:1907.03525

Let G be a reductive group with Langlands dual G'. Gaitsgory conjectured that affine Category O for G at a noncritical level should be equivalent to Whittaker D-modules on the affine flag variety of G' at the dual level. We will provide motivation and background for this conjecture, which is some form of geometric Satake for quantum groups. We have proven this conjecture when the level is appropriately integral with Justin Campbell, and the general case is work in progress with Sam Raskin. Depending on time and interest, we will also indicate its relation to a Beilinson--Bernstein style localization theorem for W-algebras, obtained jointly with Sam Raskin, and expected consequences thereof.

Let beta be a positive braid on n strands. Using microlocal techniques, Shende-Treumann-Zaslow showed the following are proportional up to a q-factor: (1) the coefficient of the highest a-power in the HOMFLY series of beta; (2) the coefficient of 1 when we write beta in the standard basis of the Iwahori-Hecke algebra of S_n. We give a new proof that moreover generalizes this result to any Weyl group W. To do this, we introduce a new class function on the Artin braid group of W that refines the Markov trace of Jones-Ocneanu-Gomi. Its output is a sum of q-graded total Springer representations, weighted by point-counts on certain stacks over F_q. If time permits, we will show that special values of this class function are characters of virtual A_W-modules, where A_W is the rational Cherednik algebra of W.

A construction due to Hausel and Proudfoot associates to a graph a complex-analytic integrable system. In this talk I will describe a formal-algebraic analogue of their construction, and explain how one can compute the p-adic volumes of the generic Hitchin fibres in graph-theoretic terms. This is joint work with Michael McBreen.

I will talk about joint work with David Speyer where we study the singular cohomology of cluster varieties, and in particular about the "curious Lefschetz duality" that these cohomology rings satisfy. There are relations to the study of higher extension groups of Verma modules in the principal block of Category O, and to the study of rational q,t-Catalan polynomials.

Brenti proves a non-recursive formula for the Kazhdan-Lusztig polynomials of Coxeter groups by combinatorial methods. In the case of the Weyl group of a split group over a finite field, a geometric interpretation of the formula is given by Sophie Morel via the "weight-truncation" theory of perverse sheaves developed by her. With suitable modifications of Morel's proof, we generalize the geometric interpretation to the case of finite and affine partial flag varieties. We also demonstrate the result for sl_3, sl_4 and affine sl_2

In this talk I outline recent developments in representation theory and enumerative geometry inspired by 3d mirror symmetry.

For projective spaces we know many different semiorthogonal decompositions of their derived categories of coherent sheaves. It's generally expected, but not known aside from the case of P^1, that those standard decompositions are the only possible ones. For a projective plane I will show that a semiorthogonal decomposition D(P^2) = < A, B > is standard, i.e. it comes from the standard exceptional collection, under the additional assumption that either A or B contains a nonzero torsion object. I will also explain why this assumption is reasonable and what needs to be done to remove it.

It is well known that Hecke algebra is spanned by braids modulo skein relations, while the span of closed braids in the annulus modulo skein relations is isomorphic to the space of symmetric functions. I will describe a categorification of these results: the category of Soergel bimodules categories the Hecke algebra, while the annular closure corresponds to the formalism of "derived horizontal trace". Along the way, I will explicitly compute Hochschild homology of the category of Soergel bimodules.

All notions will be explained in the talk. This is a joint work in progress with Matt Hogancamp and Paul Wedrich.

In this talk, we will discuss the K-theoretic Hall algebra of a surface, which generalizes Grojnowski's and Nakajima's description of the cohomology of Hilbert schemes of points on surfaces. We will also discuss a wheel condition for the shuffle presentation of K theoretic Hall algebra, which originated from Feigin and Odesskii.

Given a quiver with potential, Kontsevich-Soibelman constructed a Hall algebra on the cohomology of the stack of representations of (Q,W). In particular cases, one recovers the Yangian of a quiver Q as defined by Maulik-Okounkov. However, for general (Q,W), the Hall algebra has nice structure properties, for example Davison-Meinhardt proved a PBW theorem for it using the decomposition theorem.

One can define a K-theoretic version of this algebra using certain categories of singularities that depend on the stack of representations of (Q,W). In particular cases, these Hall algebras are quantum affine algebras. We show that some of the structure properties in cohomology, such as the PBW theorem, can be lifted to K-theory, replacing the use of the decomposition theorem with semi-orthogonal decompositions.

The Affine Grassmannian is an ind-scheme associated to a reductive group G. It has a cell structure similar to the one in the usual Grassmannian. Transversal slices to these cells give an interesting family of Poisson varieties. Some of them admit a smooth symplectic resolution and have an interesting geometry related to the representation theory of the Langlands dual group. We will focus on equivariant cohomology of such resolutions and will show how the trigonometric Knizhnik-Zamolodchikov equation arises as a quantum differential equation in this setting.