Meeting Time: Fridays, 3:00 PM  4:30 PM  Location: Virtual through Zoom
Coordinators: Pavel Etingof, Victor Kac, and Andrei Negut
Date and Time  Speaker  

February 19, 3:00 PM  4:30 PM  Dennis Gaitsgory (Harvard) 
KazhdanLusztig equivalence via screening operators The talk will revisit the "screening charges", which are some particular elements in Wakimoto modules, first discovered in the works of Feigin and Frenkel. We will explain their categorical and factorization interpretations, and show how they can be used to construct an explicit functor from the category of modules over the (big) quantum group to the KazhdanLusztig category of modules over the affine KacMoody algebra. 
February 26, 3:00 PM  4:30 PM  Olivier Schiffmann (CNRS, ParisSud) 
Cohomological Hall algebras associated to ADE surface singularities To a (reasonable) CY category C of global dimension two one can attach an associative algebra its cohomological Hall algebra (COHA) which is an algebra structure on the BorelMoore homology of the stack of objects in C. In examples related to quivers (i.e. when C is the category of representations of the preprojective algebra of a quiver Q) this yields (positive halves) of KacMoody Yangians. In ongoing joint work with Diaconescu, Sala and Vasserot, we consider the case of the category of coherent sheaves supported on the exceptional locus of a Kleinian surface singularity. This is related to the above quiver case by a '2dCOHA' version of Cramer's theorem relating the (usual) Hall algebras of two hereditary categories which are derived equivalent. 
March 5, 3:00 PM  4:30 PM  Allen Knutson (Cornell) 
Bruhat cells, subword complexes, and stratified atlases The finitedimensional Bruhat cells in KacMoody flag varieties are stratified by their intersections with the finitecodimensional Schubert varieties. This stratification has many excellent combinatorial and geometric qualities: it is generated (in a precise sense) by the hypersurface complementary to the open stratum, and every stratum is normal with anticanonical boundary. I'll trace this to the fact that the hypersurface is degree n in n variables with leading term the product of variables. Using further leading term (Gr\"obner basis) technology I'll rederive standard results about Bruhat order, and deeper ones like the fact that subword complexes are balls or spheres.
With Bruhat cells firmly in place, I'll use them to put "Bruhat atlases" on famous stratified spaces. This latter work is joint with X. He and J.H. Lu, with results by Snider, Huang, KWooYong, Elek, and GalashinKarpLam. 
March 12, 3:00 PM  4:30 PM  Jose Simental Rodriguez (MPIM) 
Gieseker varieties, affine Springer fibers, and higher rank (q,t)Catalan numbers Fix coprime positive integers m and n, and a positive integer r. In earlier work with Etingof, Krylov and Losev, we defined the (m,n)Catalan number of rank r as the dimension of the unique irreducible finitedimensional representation of a quantization of the Gieseker moduli space of rank r torsion free sheaves on P^2 with fixed trivialization at infinity and second Chern class n. We related this
representation to representations of rational Cherednik algebras, where a mysterious (m,n) switch appeared. After recalling this work, I will use the geometry of affine Springer fibers to explain why this switch is not so mysterious after all, and to produce a (q,t)deformation of the higher rank Catalan numbers. This talk is based on joint works with various subsets of {P. Etingof, E. Gorsky, V. Krylov, I. Losev and M.
Vazirani}. 
March 19, 3:00 PM  4:30 PM  Andrei Negut (MIT) 
Lyndon words and quantum loop groups Shuffle algebras provide combinatorial models for U_q(n), i.e. half of a quantum group. We define a loop version of this construction, i.e. a combinatorial model for U_q(Ln), and connect it with Enriquez' degeneration A of the elliptic algebras of FeiginOdesskii. Our techniques involve constructing a PBW basis of U_q(Ln) indexed by standard Lyndon words, generalizing the work of LalondeRam, Leclerc and Rosso in the U_q(n) case. As an application, we prove a conjecture that describes the image of the embedding U_q(Ln) > A in terms of pole and wheel conditions. Joint work with Alexander Tsymbaliuk. 
March 26, 3:00 PM  4:30 PM  Andras Szenes (Geneva) 
P=W, equivariant integration and residues The P=W conjecture of de Cataldo, Hausel and Migliorini may be interpreted as a refined structure on the equivariant intersection numbers of the moduli space of Higgs bundles, which in turn, may be given the form of an elaborate enumerative identity. I will describe joint work with Simone Chiarello and Tamas Hausel, on a partial resolution of the problem in rank 2. 
April 2, 3:00 PM  4:30 PM  Edward Frenkel (Berkeley) 
Analytic version of the Langlands correspondence and quantum integrable systems The Langlands correspondence for complex curves has been traditionally
formulated in terms of sheaves rather than functions. Together with
Pavel Etingof and David Kazhdan (arXiv:1908.09677, arXiv:2103.01509), we
have formulated a functiontheoretic (or analytic) version as a spectral
problem for an algebra of commuting operators acting on halfdensities
on the moduli space of Gbundles over a complex algebraic curve. This
algebra is generated by the global differential operators on Bun_G and
Hecke operators. We conjecture that the joint spectrum of this algebra
(properly understood) can be identified with the set of opers for the
Langlands dual group of G whose monodromy is in the split real form (up
to conjugation).
I will start the talk with a brief introduction to the Langlands
correspondence, both geometric and analytic versions, and
relations between them. I will then talk about the analytic version in
terms of a quantum integrable system obtained by "doubling" the
celebrated quantum Hitchin system (also known as the Gaudin system in
genus 0), i.e. combining the holomorphic and antiholomorphic degrees of
freedom. 
April 9, 3:00 PM  4:30 PM  Vera Serganova (Berkeley) 
On representations of the Lie algebra of polynomial vector fields Let W(n) denote the Lie algebra of polynomial vector fields over complex numbers in n indeterminates and H be its maximal toral subalgebra. A W(n)module M is a weight module if it is semisimple over H and has finite weight multiplicities. I will report on our recent work with D. Grantcharov in which we classify simple weight W(n)modules. This classification is based on previous work of many people dating back to 70s and in the first part of the talk I will review some of these results. At the end of the day all simple weight W(n)modules are generalized tensor modules (or explicitly defined submodules of tensor modules). Tensor modules first appeared in the work of A. Rudakov, they can be described as sections of vector bundles in affine space. Generalized tensor modules appear as pullbacks from a map of U(W(n)) to the tensor product of the Weyl algebra D(n) and U(gl(n)). 
April 16, 10:00 AM  11:30 AM (special time) 
Tomoyuki Arakawa (RIMS) 
Urod algebras and Translation of Walgebras In 2016 Bershtein, Feigin and Litvinov introduced the Urod algebra, which gives a representation theoretic interpretation of the celebrated NakajimaYoshioka blowup equations on Nekrasov partition functions in the case that the sheaves are of rank two. Urod algebras also play an important role in the recent work of Feigin and Gukov on VOA[M_4]. In this talk we will introduce higher rank Urod algebras. This is done by constructing translation functors for affine Walgebras. This is a joint work with Thomas Creutzig and Boris Feigin. 
April 23, 3:00 PM  4:30 PM  Sasha Goncharov (Yale) 
Geometric quantization of cluster varieties and representation theory I will introduce cluster geometric quantization of a cluster Poisson variety X. It depends on an arbitrary rational Planck constant h = r/s, and produces an analytic vector bundle E with a relative connection on a gerb G on (a cover of) X, with the following features:

April 30, 9:00 AM  10:30 AM (speical time) 
Hiraku Nakajima (IPMU) 
Geometric Satake for affine Lie algebras I will first review BravermanFinkelberg's (partly with myself) geometric Satake correspondence conjecture for KacMoody Lie algebras via Coulomb branches of quiver gauge theories. Most of the statements were proved in affine type A, viewing Coulomb branch as quiver varieties and use the levelrank duality. Then I would like to spend most of my time explaining one remaining statement. It is a description of the intersection cohomology as a graded vector space, which is given in terms of BrylinskiKostant filtration in the usual geometric Satake. In view of works for ArkhipovBezrukavnikovGinzburg and GinzburgRiche, we regard this problem as a Coulomb branch type construction of the cotangent bundle of a KacMoody flag variety and its quantization. We briefly explain how we show this description in affine type A. (This is a joint work with Dinakar Muthiah.) 
May 7, 3:00 PM  4:30 PM  Ivan Losev (Yale) 
BeilinsonBernstein localization theorem revisited The BeilinsonBernstein localization theorem is one of the central results in Lie representation theory. It relates representations of semisimple Lie algebras to Dmodules on flag varieties. In this talk I will discuss a new proof of this theorem due to myself. While complicated, this proof generalizes to many other situations of quantizations of symplectic resolutions. 
May 14, 3:00 PM  4:30 PM  Catharina Stroppel (Bonn) 
From diagram algebras via Ringel duality to highest weight categories Diagram algebras (like TemperleyLieb algebras, Brauer algebras,skein algebras...) can often be constructed as centraliser in some SchurWeyl duality. In this talk I will obtain them from a Ringel duality setup which will be in fact realised as a duality between categories of comodules and categories of modules constructed via some Coend construction. Based on explicit examples (including the above diagram algebras as well as diagrammatical Hecke categories or examples from Lie theory) I will give the definition of algebras with a triangular decomposition and indicate the corresponding highest weight theory. Finally the connection to classical highest weight categories and a similar notion used in geometric representation theory will be sketched. 