MIT Infinite Dimensional Algebra Seminar (Spring 2021)

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 Kazhdan-Lusztig category of modules over the affine Kac-Moody algebra.

Lecture Notes

The finite-dimensional Bruhat cells in Kac-Moody flag varieties are stratified by their intersections with the finite-codimensional 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, K-Woo-Yong, Elek, and Galashin-Karp-Lam.

Lecture Notes

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 finite-dimensional 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}.

Lecture Notes

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 Feigin-Odesskii. Our techniques involve constructing a PBW basis of U_q(Ln) indexed by standard Lyndon words, generalizing the work of Lalonde-Ram, 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.

Lecture Notes

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.

Lecture Notes

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 function-theoretic (or analytic) version as a spectral problem for an algebra of commuting operators acting on half-densities on the moduli space of G-bundles 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 anti-holomorphic degrees of freedom.

Lecture Notes

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 70-s 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 pull-backs from a map of U(W(n)) to the tensor product of the Weyl algebra D(n) and U(gl(n)).

In 2016 Bershtein, Feigin and Litvinov introduced the Urod algebra, which gives a representation theoretic interpretation of the celebrated Nakajima-Yoshioka 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 W-algebras. This is a joint work with Thomas Creutzig and Boris Feigin.

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:

i) dim(E) = s^n, where n is the dimension of the generic symplectic leaf.
ii) The slope of E = h times (the relative symplectic form).
iii) The cluster modular group of X acts by automorphisms of E.
iv) There is a sheaf of Azumaya algebras quantising X, acting by automorphisms of E.

I will explain an application of this construction to the representation theory of the DeConcini-Kac quantum groups at roots of unity. It provides an analog of the braided monoidal category for the generic representations of the DeConcini-Kac quantum group. Note that the naive tensor category of the latter can not be braided.

I will first review Braverman-Finkelberg's (partly with myself) geometric Satake correspondence conjecture for Kac-Moody 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 level-rank 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 Brylinski-Kostant filtration in the usual geometric Satake. In view of works for Arkhipov-Bezrukavnikov-Ginzburg and Ginzburg-Riche, we regard this problem as a Coulomb branch type construction of the cotangent bundle of a Kac-Moody 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.)

Lecture Notes

The Beilinson-Bernstein localization theorem is one of the central results in Lie representation theory. It relates representations of semisimple Lie algebras to D-modules 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.

Lecture Notes

Diagram algebras (like Temperley-Lieb algebras, Brauer algebras,skein algebras...) can often be constructed as centraliser in some Schur-Weyl 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.

Lecture Notes