MIT Infinite Dimensional Algebra Seminar (Spring 2013)

I will give an introduction to the works of Deligne, who constructed extrapolations$O(n,C)$ and $Sp(2n,C)$) to complex values of $n$. Then I will describe how to generalize this to $GL_n(F_q)$, real groups (i.e., Harish-Chandra modules), Lie superalgebras, degenerate affine and double affine Hecke algebras, current algebras, Yangians, and other representation theoretical settings. Finally, I will describe some open problems. This will be a beginning of a student seminar on this subject, which will operate on Tuesdays.

We define a notion of a tensor product of categorifications of irreducible representations of a Kac-Moody algebra and prove that such a tensor product is unique. In my talk I will concentrate on the case when the algebra is sl_n. The talk is based on a joint work with Ben Webster (in preparation).

In my talk I present a new approach to the local Langlands correspondence, developed recently joinly with A.Gerasimov and D.Lebedev. The approach is based on a novel interpretation of the Archimedean Langlands correspondence in terms of the mirror symmetry beween a pair topological sigma-models an a two-dimensional disk, which reproduce local Archimedean L-factions and the Whittaker functions as the correlation functions.

In the first part of my talk I recall basic facts on G-Whittaker functions for a reductive quasisplit G=G(F) over a local field F and introduce their q-deformations. In the case of non-Archimedean F the local Langlands correspondence provides an explicit formula, identifying the G(F)-Whittaker function with a character of a (complex) dual group. This formula was conjectured by Langlands in 1967 and then proved by Shintani and Casselman and Shalika. I propose a q-analog of this explicit formula for the q-deformed Whittaker funciton, which interpolates between the Archimedean and non-Archimedean cases, when specializing the deformation parameter q. At the end of my talk I explain the Archimedean counterpart of the Langlands-Shintani formula and its relation to the mirror symmetry in two-dimensional topological field theories.

Finite type knot invariants are those invariants vanishing on the n-th piece of some natural filtration on the space of knots. This notion was introduced by Vassiliev and it turns out that most of known numerical invariants are of finite type. Kontsevich proved the existence of a "universal" invariant, taking its values in some combinatorial space, of which every finite type invariant is a specialization. This result involves some complicated integrals, but can be made combinatorial using the theory of Drinfeld associators. We will review this construction and explain why the naive generalization of this theory for knot in thickened surfaces fails. We will suggest a general way of overcoming this obstruction, and prove an analog of Kontsevich theorem in this framework for the case M=C^*, i.e. for knots in a solid torus. Time permitting, we will give an explicit construction of specializations of our invariant using quantum groups.

We study a family of abelian categories $O_{c, t}$ depending on complex parameters $c, t$ which are interpolations of the $O$-category for the rational Cherednik algebra $H_c(t)$ of type $A$, where $t$ is a positive integer. This interpolation is based on Deligne's construction of the category Rep(S_t), which was discussed in Deligne Categories Seminar.

The symmetric q-Whittaker function attracts a lot of attention now. Its nonsymmetric generalization and the related theory of q-Toda-Dunkl operators was an unexpected development, quite involved even for A1. I will discuss our last paper with Dan Orr devoted to this theory for arbitrary reduced root systems (the twisted setting). Geometrically, the nonsymmetric Whittaker function we introduced is a quadratic-type generating function of the level-one Demazure characters for all (not only dominant) weights. The new technique of W-spinors is used, which is expected to influence classical real and p-adic theory of Whittaker functions and find applications in the theory of affine flag varieties.

I will also touch upon a surprising connection we found to the PBW-filtration (an ongoing project with Evgeny Feigin). The latter is closely related to the Kostant q-partition function, though in a way different from that for the Lusztig's q-analogs of weight multiplicities and the BK-filtration. We bumped into the Kostant q-partition function when calculating the extremal q-powers for the so-called E-dag-polynomials, dual to the nonsymmetric q-Hermite ones. This resulted in a new approach to the PBW-filtrartion (E.Feigin, Fourier, Littelmann), though only for extremal weights so far, which will be discussed in the second half of the talk.