Geometric Representation Theory

I will describe a joint project with Galyna Dobrovolska and Ivan Losev. The goal is to prove an equivalence between certain noncommutative resolutions of the symmetric power of the complex plane and a category of perverse sheaves on the moduli space of stable pairs on an elliptic curve. Various ingredients come from geometric Langlands duality, from the theory of elliptic Hall algebras, from representation theory of rational DAHA in positive characteristic and from the theory of stabilities a la Bridgeland. The conjectural equivalence relates that type of stability on the coherent side to the classical GIT stability on the constructible side

In this talk, I will explain how to use Bridgeland stability to study the birational geometry of moduli spaces of sheaves on the projective plane. I will show that wall-crossing naturally provides smooth projective birational models of these moduli spaces, and induces the MMP for them. Moreover a numerical criterion of potential walls being actual walls will be given, and used to compute several cones of these moduli spaces. If time permits, I will also discuss a generalization to non-commutative projective planes, and an application to the study of linear series on these moduli. This talk is based on a series of joint works with Chunyi Li.

Wall-crossing functors for the rational Cherednik algebra were introduced by Bezrukavnikov and Loseu, and were subsequently shown by Loseu to be described by the Mullineux involution in the theory of modular representations of the symmetric group. I will talk about some partial results on a conjecture of Bezrukavnikov on combinatorial wall-crossing for the rational Cherednik algebra in characteristic p. This is joint work with Panagiotis Dimakis and Guangyi Yue.

In joint work with Buryak, Pandharipande and Tessler (in preparation), we define equivariant stationary descendent integrals on the moduli of stable maps from surfaces with boundary to (CP^1 , RP^1). For stable maps of the disk, the definition is geometric and we prove a fixed-point formula involving contributions from all the corner strata. We use this fixed-point formula to give a closed formula for the integrals in this case.

We conjecture that the higher genus theory exists and give an explicit fixed-point formula for it. We show that this formula behaves as expected in the closed and disk-map sector, and satisfies a domain decomposition property. We consider open branched covers and prove they satisfy analogous properties. This implies a Gromov-Witten Hurwitz correspondence extending part of the work of Okounkov and Pandharipande from the closed to the open setting.

In the talk I'll discuss some of these ideas. I'll try to make the talk as self-contained as possible, and give examples.

We will describe the computation of the monodromy group of compact hyperkahler varieties deformation equivalent to the generalized Kummer variety associated to an abelian surface. As an application we will explain O'Grady's recent observation that complete families of abelian fourfolds of Weil type (precisely those abelian fourfolds for which the Hodge conjecture is open) arise as intermediate Jacobians of generalized Kummers.

Mukai, Orlov, and Polishchuk have shown that the group of autoequivalences of the derived category of coherent sheaves on a g dimensional abelian variety acts on its cohomology via the natural action of Spin(4g). This action plays the role of a symmetry group for nice moduli spaces of stable sheaves on the abelian variety. The generalized Kummers are fibers of the Albanese map from moduli spaces of stable sheaves on an abelian surface (g=2). In this case an integral version of triality for Spin(8) is instrumental in computing the monodromy group for generalized Kummers.

The theory of factorization algebras developed by Kevin Costello and Owen Gwilliam describes the algebraic structure of observables in quantum field theories. When the space-time manifold has complex dimension 1, a certain class of factorization algebras corresponds to vertex algebras. I will give a sketch of this theory and describe a construction of factorization algebras on Riemann surfaces from holomorphic fibrations. When the fiber is a torus, the corresponding factorization algebra gives a vertex realization of a toroidal algebra. This is joint work with Jackson Walters.

Consider the tilings of an oriented surface by triangles, or squares, or hexagons, up to combinatorial equivalence. The combinatorial curvature of a vertex is 6, 4, or 3 minus the number of adjacent polygons, respectively. Tilings are naturally stratified into all such having the same set of non-zero curvatures. We outline a proof that for squares and hexagons, the generating function for the number of tilings in a fixed stratum lies in a ring of quasi-modular forms of specified level and weight. First, we rephrase the problem in terms of Hurwitz theory of an elliptic orbifold---a quotient of the plane by an orientation-preserving wallpaper group. In turn, we produce a formula for the number of tilings in terms of characters of the symmetric group. Generalizing techniques pioneered by Eskin and Okounkov, who studied the pillowcase orbifold, we express the generating function for a stratum in terms of the q-trace of an operator acting on Fock space. The key step is to compute the trace in a different basis to express it as an infinite product, then apply the Jacobi triple product formula to conclude quasi-modularity.

The relaxed highest weight representations introduced by Feigin, Semikhatov, and Tipunin are a class of representations of the Lie algebra affine sl2 , which do not have a highest (or lowest) weight. We generalize this notion to an arbitrary affine Kac-Moody algebra g and realize induced g-modules of this type and their duals as global sections of twisted D-modules on the Kashiwara flag scheme associated to g. The D-modules that appear in our construction are direct images from subschemes given by the intersection of finite dimensional Schubert cells with their translate by a simple reflection. Besides the twist, they depend on a complex number describing the monodromy of the local systems we construct on these intersections. These results describe for the first time explicit non-highest weight g-modules as global sections on the Kashiwara flag scheme and extend results of Kashiwara-Tanisaki to the case of relaxed highest weight representations. This is based on arxiv:1607.06342 [math.RT]

Let G be a complex semisimple algebraic group of adjoint type and bar{G} the wonderful compactification. We show that the closure in G of the centralizer G^e of a regular nilpotent e in Lie(G) is isomorphic to the Peterson variety. We generalize this result to show that for any regular x in Lie(G), the closure of the centralizer G^x in G is isomorphic to the closure of a general G^x -orbit in the flag variety. We consider the family of all such centralizer closures, which is a partial compactification of the universal centralizer. We show that it has a natural log-symplectic Poisson structure that extends the usual symplectic structure on the universal centralizer.

The integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface arise in enumerative problems. We explain an approach to the K-theoretic analogs of such expressions. Namely, we explain how to compute the K-theoretic Donaldson-Thomas partition function of a Calabi-Yau threefold, and show how this computation implies certain symmetries of generating functions of equivariant Euler characteristics of tautological classes on the Hilbert scheme.

In this talk I will define the quantum K-theory of Nakajima quiver varieties and show its connection to representation theory of quantum groups and quantum integrable systems on the examples of the Grassmannian and the flag variety. In particular, the Baxter operator will be identified with operators of quantum multiplication by quantum tautological classes via Bethe equations. Quantum tautological classes will also be constructed and, time permitting, an explicit universal combinatorial formula for them will be shown. Based on joint works with P.Koroteev, A.Smirnov and A.Zeitlin

Abstract: Cohomological Hall algebras associated with preprojective algebras of quivers play a preeminent role in algebraic geometry and representation theory. For example, if the quiver is the 1-loop quiver, the corresponding CoHA is the Maulik-Okounkov affine Yangian of gl(1). It acts on the equivariant cohomology of Hilbert schemes of points on the complex affine place ("extending" previous results of Nakajima, Grojnowski, Vasserot, etc, for action of Heisenberg algebras) and moduli spaces of framed sheaves on the complex projective plane. In the present talk, I will follow the recipe of "replacing quivers with curves" and I will end up by introducing cohomological Hall algebras associated with the stack of Higgs sheaves on a smooth projective complex curve. (This is a joint work with Olivier Schiffmann)

The main part of the talk is joint work with S. Arkhipov. We establish an equivalence between the derived category of coherent sheaves on a certain kind of DG-schemes and the absolute derived category of matrix factorizations. One example of such is the derived category of coherent sheaves on the Steinberg variety, which has a braid group action constructed by Bezrukavnikov and Riche. We transfer this action to the corresponding matrix factorization category. In the last part of the talk I will report on joint work in progress with R. Bezrukavnikov relating this to link invariants