MIT Infinite Dimensional Algebra Seminar (Spring 2021)

Meeting Time: Fridays, 3:00 PM - 4:30 PM | Location: Virtual through Zoom; please contact Andrei Negut to be placed on the mailing list and to receive Zoom link and password.

Contact: Pavel Etingof, Victor Kac, and Andrei Negut

Schedule of Talks

Date and Time Speaker
February 19, 3:00 PM - 4:30 PM Dennis Gaitsgory (Harvard)

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

Lecture Notes

February 26, 3:00 PM - 4:30 PM Olivier Schiffmann (CNRS, Paris-Sud)

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 Borel-Moore 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 Kac-Moody 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 '2d-COHA' version of Cramer's theorem relating the (usual) Hall algebras of two hereditary categories which are derived equivalent.

Lecture Notes

March 5, 3:00 PM - 4:30 PM Allen Knutson (Cornell)

Bruhat cells, subword complexes, and stratified atlases

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

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 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

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 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

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.

Lecture Notes

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 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

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 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)).

April 16, 10:00 AM - 11:30 AM (special time)

Tomoyuki Arakawa (RIMS)

Urod algebras and Translation of W-algebras

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.

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:

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.

April 30, 9:00 AM - 10:30 AM (speical time)

Hiraku Nakajima (IPMU)

Geometric Satake for affine Lie algebras

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

May 7, 3:00 PM - 4:30 PM Ivan Losev (Yale)

Beilinson-Bernstein localization theorem revisited

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

May 14, 3:00 PM - 4:30 PM Catharina Stroppel (Bonn)

TBD

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