Geometric Representation Theory

Wreath Macdonald polynomials were defined by Haiman as generalizations of transformed Macdonald polynomials from the symmetric groups to wreath products with cyclic groups of order m. In a sense, their definition was given in the hope that they would correspond to K-theoretic torus-fixed points of cyclic quiver varieties, much like how Haiman's proof of Macdonald positivity assigns Macdonald polynomials to torus-fixed points of Hilbert schemes of points on the plane. This hope was realized by Bezrukavnikov and Finkelberg, and the subject has been relatively untouched until now. I will present a first result exploring possible ties to integrable systems. Using work of Frenkel, Jing, and Wang, we can situate the wreath Macdonald polynomials in the vertex representation of the quantum toroidal algebra of sl_m. I will present a proof that in this setting, the wreath Macdonald polynomials diagonalize the horizontal Heisenberg subalgebra of the quantum toroidal algebra.

Classical work by Thurston in the theory of surfaces gives symplectic co-ordinate charts on Teichmüller space, associated to quadratic differentials. Motivated by wall crossing in 4d field theories Gaiotto, Moore and Neitzke defined a generalization of these charts for SL(n, C) character varieties of non-compact Riemann surfaces. We will describe out extension of this construction to an arbitrary reductive algebraic group G, and discuss ongoing work on our approach to proving these charts are holomorphic symplectic.

Let M be a manifold with non-vanishing vectorfield. The homology of the space of loops in M carries a natural Lie bialgebra structure described by Sullivan as string topology operations. If M is a surface, these operations where originally defined by Goldman and Turaev. We study formal descriptions of these Lie bialgebras. More precisely, for surfaces these Lie bialgebras are formal in the sense that they are isomorphic (after completion) to their algebraic analogues (Schedler's necklace Lie bialgebras) built from the homology of the surface. For higher dimensional manifolds we give a similar description that turns out to depend on the Chern-Simons partition function.

This talk is based on joint work with A. Alekseev, N. Kawazumi, Y. Kuno and T. Willwacher.

We introduce the notion of R-maps which further generalizes the hybrid model by considering pre-stable maps to possibly non-GIT quotients together with sections of certain spin bundles, usually called the p-fields. The moduli of R-maps are in general non-compact. When the target of R-maps is equipped with a super-potential W with compact critical locus, using Kiem-Li cosection localization it has been proved by many authors in various settings that the virtual cycle of R-maps can be represented by the cosection localized virtual cycle which is supported on the proper locus consisting of R-maps in the critical locus of W. Though the moduli of R-maps is equipped with a natural torus action by scaling of the spin bundles, the non-compactness of the R-maps moduli makes such torus action less useful.

In this talk, I will introduce a logarithmic compactification of the moduli of R-maps using certain modifications of stable logarithmic maps. The logarithmic moduli space carries a canonical virtual cycle from the logarithmic deformation theory. In the presence of a super-potential with compact critical locus, it further carries a reduced virtual cycle. We prove that (1) the reduced virtual cycle of the compactification can be represented by the cosection localized virtual cycle; and (2) the difference of the canonical and reduced virtual cycles is another reduced virtual cycle supported along the logarithmic boundary. As an application, one recovers the Gromov-Witten invariants of the critical locus as the invariants of logarithmic R-maps of its ambient space in an explicit form. The latter can be calculated using the spin torus action.

This is a joint work with Felix Janda and Yongbin Ruan.

We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface. We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3. Finally, if time allows at the end, I will further report on extensions of this project to more general geometric backgrounds for which techniques in derived algebraic geometry can be used to define and compute invariants. This is based on joint works with Diaconescu-Yau, and Borisov.

I will discuss recent work on constructing small quantum groups - also known as Frobenius-Lusztig kernels - at even roots of unity. In particular, for any simple Lie algebra g and even order q, we would like to associate a corresponding finite-dimensional, factorizable, ribbon (i.e. log-modular) quasi-Hopf algebra. The main issue here is that, for g = sl2 at any even root of unity, for example, naive construction of such quantum groups produce finite tensor categories which admit no braidings. Our investigation is motivated by conjectural relations between non-rational vertex operator algebras and such log-modular quantum groups, which I will also discuss.

I will describe how the (co)homology of "diagonal" affine Springer fibers in the affine Grassmannian relates to projective coordinate rings of certain varieties. In the G=GL(n) case, we recover Haiman's isospectral Hilbert scheme on the surface C* x C. Time permitting, I will explain how this construction relates to Cherednik algebras and knot homology.

Toda lattices play a distinguished role in both the classical and modern theories of completely integrable systems, and they are fruitfully studied at the interface of symplectic geometry and representation theory. One crucial aspect of this study is Kostant's Lie-theoretic realization of the (open) Toda lattice, which one sometimes calls the Kostant-Toda lattice. This construction invokes Kostant's prior works on invariant theory, especially his results on regular Slodowy slices and the structure of the adjoint quotient.

I will discuss invariant-theoretic aspects of the Kostant-Toda lattice, emphasizing two recent constructions. The first is a partial compactification of the Kostant-Toda lattice by means of Hessenberg varieties, Slodowy slices, and Mishchenko-Fomenko algebras, and it represents joint work with Hiraku Abe. The second construction is a Toda-type integrable system on the universal centralizer, a hyperkahler manifold studied by Balibanu, Ginzburg, and others.

Consider the derived category of sheaves on a reductive group G equivariant under left and right actions of the maximal unipotent U and with prescribed monodromy under the left and right actions of the maximal torus. This category carries a monoidal structure under convolution. We show that this monoidal category can essentially be identified with the usual Hecke category (trivial torus monodromy) for an endoscopic group H of G, which is a reductive group sharing a maximal torus with G but is not necessarily a subgroup of G. Joint work with G.Lusztig.

The Yangian associated to a simple Lie algebra, is a Hopf algebra which comes equipped with a universal R-matrix. This R-matrix, as introduced by Drinfeld in 1985, is a formal series in 1/u, with values in a two-fold tensor product of the Yangian.

In this talk, I will present a proof of the following "folk-lore theorem": once evaluated on a pair of finite-dimensional representations, Drinfeld's formal R-matrix is the asymptotic expansion of two different meromorphic functions of one complex variable, related by a unitarity condition. I will also explain how to factorize these two meromorphic functions as a product of Yangian intertwiner and a rational R-matrix. This factorization, however, is not natural.

This talk is based on a joint work, in progress, with V. Toledano Laredo and C. Wendlandt.

For a hyper-kahler variety equipped with a Lagrangian fibration, an increasing filtration is defined on its rational cohomology using the perverse t-structure. We will discuss the role played by this filtration in the study of the topology and geometry of hyper-kahler varieties, as well as the connection to curve counting invariants of Calabi-Yau 3-folds. In particular, we will discuss some recent progress on the P=W conjecture for Hitchin systems, and its compact analog for Lagrangian fibrations. Based on joint work with Qizheng Yin and Zili Zhang.

The Grothendieck ring of varieties is the target of a rich invariant associated to any algebraic variety which witnesses the interplay between geometric, topological and arithmetic properties of the variety. The motivic Hilbert zeta function is the generating series for classes in this ring associated to a certain compactification of the unordered configuration space, the Hilbert scheme of points, of a variety. In this talk I will discuss the behavior of the motivic Hilbert zeta function of a reduced curve with arbitrary singularities. For planar singularities, there is a large body of work detailing beautiful connections with enumerative geometry, representation theory and topology. I will discuss some possible extensions of this picture to non-planar curves.

Let G be a split reductive p-adic group. In the Iwahori-invariants of an unramified principal series representation of G, there are two bases, one of which is the so-called Casselman basis. In this talk, we will prove a conjecture of Bump--Nakasuji--Naruse about certain transition matrix between these two bases. The idea of the proof is to use the two geometric realizations of the affine Hecke algebra, and relate the Iwahori invariants to Maulik--Okounkov's stable envelopes and Brasselet--Schurmann--Yokura's motivic Chern classes for the Langlands dual groups. This is based on joint work with P. Aluffi, L. Mihalcea and J. Schurmann.

I will discuss joint work with Joel Kamnitzer and Nick Proudfoot that relates the quantum connection of a symplectic resolution (at certain special values of the parameters) with the character theory of the quantized symplectic dual resolution.

Symplectic reflection algebras were introduced by Etingof and Ginzurg in 2001. One motivation to study them is that they are closely related to quantizations of symplectic quotient singularities. In particular, they fit into a broader class of quantized symplectic singularities -- a class of associative algebras whose representation theory was studied very extensively recently by Bezrukavnikov, Kamnitzer, Okounkov, Webster, the speaker and many others. One strategy used to study the representation theory is to look at symplectic resolutions for symplectic singularities where they exist. However, such resolutions do not exist in general. What does exist always is a nice Poisson partial resolution called a Q-factorial terminalization, which is a more complicated object than a symplectic resolution as it is still singular.

In this talk I will discuss some results about the representation theory of symplectic reflection algebras that can be proved using the quantizations of Q-factorial terminalizations: derived equivalence of categories of modules under an integral shift of parameter, which is a more general version of Rouqier's conjecture from 2004, and a generalized Benrstein inequality conjectured by Etingof and Ginzburg around the same time. I will also state some open questions regarding the representation theory of quantizations of Q-factorial terminalizations. The talk is based on arXiv:1704.05144. No preliminary knoweledge of symplectic reflection algebras and of Q-factorial terminalizations will be assumed.

Taking cohomology of moduli spaces has typically provided constructions of algebras of symmetries in integrable systems, and the typical trichotomy of integrable systems teaches us to expect a natural third theory to complete the sequence that begins with ordinary cohomology and K-theory. This theory is elliptic cohomology, which has historically been hard to work and compute with due to the lack of good geometric representatives. In this talk, I'll explain how such representatives arise in equivariant elliptic cohomology, at least after tensoring with the complex numbers, in joint work with D. Berwick-Evans. As time permits, I'll indicate forthcoming extensions of the theory.

In this talk I will discuss a "curved" version of the Hecke category, introduced in joint work with Gorsky, whose objects can be described as certain curved complexes of Soergel bimodules. Our main result uses a certain link splitting property in the curved Hecke category to explicitly compute the Khovanov-Rozansky homologies of the (n,nk) torus links, both in the curved and uncurved settings. Summing over all k yields an explicit graded algebra considered by Haiman in his study of Hilbert schemes. Combining with an idea of Gorsky-Negut-Rasmussen, we obtain a functor from the (curved or uncurved) Hecke category to sheaves on the appropriate Hilbert scheme.

The Kazhdan-Lusztig equivalence says that for a negative level -\kappa, the category KL(G,-\kappa) of G(O)-integrable modules over the Kac-Moody algebra \hat{g} at level -\kappa is equivalent to the category of finite-dimensional modules over Lusztig's quantum group U_q(G). In this talk we will explain a (conjectural) extension of this equivalence, where on the Kac-Moody side, instead of G(O)-integrability we impose Iwahori integrability. We will see that on the quantum group side we will have to consider a hybrid between Lusztig's and De Concini-Kac quantum groups.

Via the geometric Satake correspondence, the Mirkovic-Vilonen cycles give bases for representations of semisimple Lie algebras. Similarly, by work of Lusztig, generic preprojective algebra modules give bases for these representations as well. It is a long-standing open problem to compare these bases. We will explain a new geometric way to make this comparison. As an application, we will show that these bases do not agree in SL_6. We will also explain the relationship with the work of Braverman-Finkelberg-Nakajima on Coulomb branches.

In this talk, we discuss a (quite lengthy) purely combinatorial formula for the weight polynomial of a general affine Springer fiber for SL_n. The formula is obtained by computing the corresponding p-adic orbital integral. Without assuming any background on harmonic analysis of p-adic groups, we will sketch the method, which essentially computes the so-called Shalika germs for GL_n. After that we will pose questions about possible geometric meaning of the formula and of the Shalika germs.

In this talk, we introduce multivariable generalizations of familiar infinite dimensional Lie algebras with a focus on two familiar examples: the Kac-Moody and the Virasoro Lie algebras. Our construction is geometric, and uses the theory of factorization algebras defined on higher dimensional complex manifolds. We will characterize central extensions using knowledge of certain Gelfand-Fuks cohomologies. In addition, we will show evidence for a higher vertex algebra structure on the respective vacuum modules using a higher dimensional analog of the operator product expansion coming from factorization. Throughout the talk we will remark on the appearance of these Lie algebras as symmetries of holomorphic quantum field theories in arbitrary dimensions.

The q-version of a Toda system associated with any Lie algebra was introduced independently by Etingof and Sevostyanov in 1999. In this talk, we shall discuss the generalization of this construction which naturally produces a family of 3^{rk(g)-1} similar integrable systems. One of the key ingredients in the proof is played by the fermionic formula for the J-factors (defined as pairing of two Whittaker vectors in Verma modules), due to Feigin-Feigin-Jimbo-Miwa-Mukhin. In types A and C, our construction admits an alternative presentation via local Lax matrices, similar to the classical construction of Faddeev-Takhtajan for the classical type A Toda system. Finally, we shall discuss the geometric interpretation of Whittaker vectors in type A. This talk is based on the joint work with M. Finkelberg and R. Gonin