MIT Infinite Dimensional Algebra Seminar (Spring 2019)

In this talk we will explain the construction of (several versions of the) FLE, and equivalence between metaplectic Whittaker sheaves and quantum groups for a group G and modules for the Langlands-dual quantum group. The equivalence goes through relating both categories to that of Factorizable Sheaves.

Chern-Simons theory is a gauge theory describing the moduli space of flat connections on 3-manifolds. When the 3-manifold has a boundary endowed with a complex structure, the theory admits natural boundary conditions which are related to some familiar vertex algebras. A well-studied example of such a correspondence is between the operators of the chiral Wess-Zumino-Witten (WZW) model, described by the affine Kac-Moody vertex algebra, and Chern-Simons theory. In this talk, we propose a general correspondence between a class of 3d topological field theories and vertex algebras using the language of factorization algebras and factorization categories. In addition to the standard CS/WZW example, we will see how the sheaf of chiral differential operators exists as the boundary chiral algebra of a certain 3d TFT labeled by a Courant algebroid. This is joint work with Pavel Safronov.

Let g be the Lie algebra of a semisimple algebraic group over an algebraically closed field of positive characteristic. In a joint work with Ivan Mirkovic we have established a relation between representations of g and canonical bases in homology of a Springer fiber predicted by Lusztig. In contrast with classical Kazhdan-Lusztig theory, the definition of a canonical basis here does not give a concrete way to compute it. I will report on a current project with Ivan Losev where computing the basis is reduced to knowing affine Kazhdan-Lusztig polynomials. This yields (reasonably) explicit formulas for dimensions of irreducible representations of g.

In this talk, I will first recall the Stokes matrices of differential equations with irregular singularities. I will then compute the WKB approximation of the Stokes matrices in the simplest case, and explore its relation with Gelfand-Zeitlin systems, total positivity of matrices, and De Concini-Kac-Procesi braid group actions. This is the first step towards the understanding of the relation between WKB analysis and integrable systems proposed by Anton Alekseev, there are many problems remianed to be understood.

We study a certain category of bimodules over a filtered algebra quantizing the algebra of functions on a conical symplectic singularity. The bimodules we care about are so called Harish-Chandra bimodules. This notion first appeared in the case of universal enveloping algebras of semisimple Lie algebras in the work of Harish-Chandra on representations of the corresponding complex Lie groups. Since then it was generalized to filtered quantizations of algebras of functions on affine Poisson varieties. The goal of this talk is to explain a classification of the simple Harish-Chandra bimodules with full support over quantizations of conical symplectic singularities (that have no slices of type E_8). We will see that these irreducible bimodules are in one-to-one correspondence with the irreducible representations of a suitable finite group. The talk is based on arXiv:1810.07625. I will not assume any preliminary knowledge of conical symplectic singularities, their quantizations etc.

Let g be an infinite-dimensional Kac-Moody algebra, and $\hat{U}$ the completion of its universal enveloping algebra with respect to highest weight representations. $\hat{U}$ is a topological Hopf algebra, which appears to have a rather unwieldy Hochschild (coalgebra) cohomology. When g is symmetrisable, Enriquez introduced a subalgebra $U$ of $\hat{U}$ which is universal among the the class of Manin triples, is large enough to define quantisation functors, and yet has a manageable Hochschild cohomology. The latter is in fact rather small since, for practical purposes, $U$ behaves like a universal enveloping algebra without primitive elements. I will give an alternative definition of $U$ using PROPs, which makes its properties far more manifest. I will then explain how this definition can be refined in various ways to accommodate in particular the 2-categorical extension of Etingof-Kazhdan quantisation which Andrea Appel and I recently obtained. This is joint work with Andrea Appel.

Following Beilinson and Drinfeld, we describe vertex algebras as Lie algebras for a certain operad of $n$-ary chiral operations. This allows us to introduce the cohomology of a vertex algebra $V$ as a Lie algebra cohomology. When $V$ is equipped with a good filtration, its associated graded is a Poisson vertex algebra. We relate the cohomology of $V$ to the variational Poisson cohomology studied previously by De Sole and Kac. This talk is based on joint work with Alberto De Sole, Reimundo Heluani, Victor Kac, and Veronica Vignoli.

Hecke categories are geometric incarnations of Hecke algebras, and they play an important role in the classification of irreducible representations of finite groups of Lie type. We consider a version of the Hecke category for the reductive group G with prescribed monodromy under the left and right actions of the maximal torus. We show that this monoidal category can essentially be identified with the usual Hecke category (with trivial torus monodromy) for an endoscopic group H of G, which is a reductive group of smaller dimension sharing a maximal torus with G but is not necessarily a subgroup of G. As a consequence, we show that character sheaves on G and on H are closely related. Joint work with G.Lusztig.

The goal of the talk is to survey recently discovered and rapidly developing interplay between solvable vertex models, symmetric functions, and probability.

It has been long understood by people at MIT and elsewhere that many highlights of Lie theory, such as the representation-theoretic theory of special functions, or the Kazhdan–Lusztig theory, have a natural extension to a much broader setting, the boundaries of which are yet to be explored. In this extension, the focus is shifting from a group G to various classes of algebraic varieties that possess the key features of T*G/B. While there are some proposal about what should replace a Lie algebra, root systems, etc., it is less clear what should be the group, or multiplicative analog of these structures. Reflecting the nature of the field, the talk will combine a review of established partial results with unsubstantiated speculations.

Azumaya algebras form a particularly nice class of algebras whose properties are determined by that of their center. In this talk I will describe Azumaya algebras which arise from deformation quantization in positive characteristic or when the quantum parameter is a root of unity. In particular, I will show that the quantization of a character variety of a closed surface (closely related to skein algebras) is Azumaya at a root of unity. This is joint work with Iordan Ganev and David Jordan.

We show that the totally nonnegative part of a partial flag variety G/P (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov. One of the main ingredients of the proof is a stratification-preserving embedding of (an open dense subset of) G/P inside the affine flag variety of G. Joint work with Steven Karp and Thomas Lam.