MIT Infinite Dimensional Algebra Seminar (Fall 2014)

I will start by explaining a result, joint with my former student Qian Lin, which links highest weight modules over affine Lie algebra at the critical level to canonical bases in homology of a Springer fiber for the dual group. The result is based on a work of Frenkel-Gaitsgory, its slight generalization would allow to compute characters of these modules. I will then talk about possible generalization of the construction of these canonical bases to affine Springer fibers and applications.

About a decade ago a surprising (and fruitful) connection between infinite dimensional Lie theory and the theory of reductive group schemes developed by Demazure and Grothendieck was found. The talk will trace this connection from its origins to the latest results.

We use the language of higher category theory to define what we call a "symmetric self-adjoint Hopf" (SSH) structure on a semisimple abelian category, which is a categorical analog of Zelevinsky's positive self-adjoint Hopf algebras. As a first result, we obtain a categorical analog of the Heisenberg double and its Fock space action, which is constructed in a canonical way from the SSH structure.

Let G be a reductive group. The tilting conjecture of Finkelberg-Mirkovic gives a geometric expression of semi-infinite cohomology of the small quantum group with coefficients in a tilting module for the big quantum group in terms of the Eisenstein series sheaf. In this talk we will show how to derive this conjecture via the technique of chiral categories. We will also give another derivation from the (still conjectural) global Geometric Langlands conjecture.

I am going to explore connections between two classes of algebras associated to complex reflection groups: Hecke algebras introduced in this generality by Broue, Malle and Rouquier, and Rational Cherednik algebras introduced by Etingof and Ginzburg. These algebras are related via the so called KZ functor between suitable categories of modules. In the first half of my talk I will introduce the algebras and the functor. In the second half I will explain various properties of the KZ functor. The talk is based on three arXiv preprints: 1409.3965, 1407.6375, 1406.7502. It is going to be reasonably self-contained.

This is a report on a joint work in progress with Michel Van den Bergh. We use the sheaf-theoretic framework of the Springer correspondence to construct a semiorthogonal decomposition of the derived category of W-equivariant coherent sheaves on the Cartan subalgebra of a complex simple Lie algebra. In the case of algebras of types A_n, B_n, C_n, G_2 and F_4, the pieces of the decomposition are numbered by the conjugacy classes in the Weyl group W and are given by derived categories of sheaves on some affine spaces. For types D_n and E_n the decomposition contains some "noncommutative" pieces. We also construct global analogs of some of these decompositions for equivariant sheaves on the powers of smooth algebraic curves.

In 1968 Tate introduced a new approach to residues on algebraic curves, based on a certain ring of operators that acts on the completion at a point of the function field of the curve. This approach was generalized to higher dimensional algebraic varieties by Beilinson in 1980. However Beilinson's paper had very few details, and his operator-theoretic construction remained cryptic for many years. Currently there is a renewed interest in the Beilinson-Tate approach to residues in higher dimensions (by Braunling, Wolfson and others). This current work also involves n-dimensional Tate spaces and is related to chiral algebras.

In this talk I will discuss my recent paper arXiv:1406.6502, with same title as the talk. I introduce a variant of Beilinson's operator-theoretic construction. I consider an n-dimensional topological local field (TLF) K, and define a ring of operators E(K) that acts on K, which I call the ring of local Beilinson-Tate operators. My definition is of an analytic nature (as opposed to the original geometric definition of Beilinson). I study various properties of the ring E(K).

In particular I show that E(K) has an n-dimensional cubical decomposition, and this gives rise to a residue functional in the style of Beilinson-Tate. I conjecture that this residue functional coincides with the residue functional that I had constructed in 1992 (itself an improved version of the residue functional of Parshin-Lomadze).

Another conjecture is that when the TLF K arises as the Beilinson completion of an algebraic variety along a maximal chain of points, then the ring of operators E(K) that I construct, with its cubical decomposition (the depends only on the TLF structure of K), coincides with the cubically decomposed ring of operators that Beilinson constructed in his original paper (and depends on the geometric input).

In the talk I will recall the necessary background material on semi-topological rings, high dimensional TLFs, the TLF residue functional and the Beilinson completion operation (all taken from Asterisque 208).

A BPS quiver is a quiver with potential whose irreducible representations can be used to characterize the spectrum of BPS excitations of a four-dimensional quantum field theory with 8 supercharges. This relation entail that the corresponding basic algebras enjoy very special properties that signal them out as extremely interesting mathematical jewels of Representation Theory. The aim of this talk is to illustrate some of the aspects of this deep and interesting interplay. In the first part of the talk, after a brief overview, I am going to discuss light subcategories and some of their properties. The latter are a generalization of the notion of regular subcategory for the representations of Euclidean algebras that follows from a very natural and Lie algebraic extension of Dlab-Ringel defect. In the second part of the talk I am going to discuss relations with number theory, singularity theory, Cluster Algebras, and cyclic subgroups of the Cremona groups in n letters.

Motivated by string theory, Aganagic and Shakirov proposed an idea of "refined" knot invariants depending on two parameters, and gave a rigorous definition of these invariants for torus knots. Conjecturally, they are related to the Poincare polynomials of Khovanov- Rozansky homology. In a joint work with Andrei Negut, we interpret these invariants as matrix elements of certain operators in the elliptic Hall algebra, also known as spherical DAHA of infinite rank. This leads to an explicit construction of sheaves on the Hilbert scheme of points such that their equivariant Euler characteristics agree with the Aganagic-Shakirov invariants (and two parameters correspond to equivariant weights). I will also discuss the recent work of Morton and Samuelson revealing a direct relation between the elliptic Hall algebra and the skein algebra of a torus.

In ongoing work with Constantin Teleman we realize the semisimple category of positive energy loop group representations as a twisted matrix factorization category. The generators are the Dirac families used in our joint work with Mike Hopkins on twisted K-theory, work I will review to begin the talk.

The purpose if the talk is to present a new geometric approach to the proof of Kazhdan-Lusztig and (partially) Jantzen conjectures for finite dimensional and affine Lie algebras. One of the advantages of this approach is that in the affine case it works uniformly in the case of positive, negative and critical level (in the latter case one gets a new result).

This is a joint work with M.Finkelberg and H.Nakajima.