Geometric Representation Theory

I'll show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A given by generators; (2) as an algebra given by generators and relations; (3) as an algebra of differential-reflection operators preserving some spaces of functions; (4) as equivariant Borel-Moore homology of a certain variety. Also, I'll define a q-deformation of this algebra, called cyclotomic DAHA. Namely, I'll give a q-deformation of each of the above four descriptions of the partially spherical rational Cherednik algebra, replacing differential operators with difference operators, degenerate DAHA with DAHA, and homology with K-theory. In addition, I'll explain that spherical cyclotomic DAHA are quantizations of certain multiplicative quiver and bow varieties, which may be interpreted as K-theoretic Coulomb branches of a framed quiver gauge theory. Finally, I'll apply cyclotomic DAHA to prove new flatness results for various kinds of spaces of q-deformed quasiinvariants. This is joint work with A. Braverman and M. Finkelberg.

In this talk, I will speak about the shifted affine quantum groups and shifted Yangians, as well as their incarnations through geometry of parabolic Laumon spaces, affine Grassmannians, additive/multiplicative slices, and Todda lattice.

The shifted Yangians were originally introduced by Brundan-Kleshchev in the gl(n) case with a dominant shift and were later generalized by Kamnitzer-Webster-Weekes-Yacobi to any simple Lie algebra with an arbitrary shift. These algebras attracted recently a new interest due to their interplay with the Coulomb branches.

In the first half of the talk, I will remind those results about shifted Yangians, while the second part will be devoted to the multiplicative analogue of this story. On the algebraic side this leads to the notion of shifted affine quantum groups, while on the geometric side we replace cohomology by K-theory and additive slices are replaced by multiplicative slices.

This is a joint project with M. Finkelberg.

One of the most intriguing conjectures arising from mirror symmetry states that various geometric invariants, some classical and some more homological in nature, agree for any two Calabi-Yau manifolds which are birationally equivalent to one another. I will discuss how new methods in equivariant geometry and geometric invariant theory have shed light on this conjecture over the past few years, leading to interesting representations of generalized braid groups as a bonus. Combined with the new theory of "Theta-stratifications" and "Theta-stability" -- a generalization of geometric invariant theory -- we have recently been able to establish the first cases of the D-equivalence conjecture for compact Calabi-Yau manifolds in dimension >3.

Strict semistability or non-semisimplicity in the sense of representation theory manifests itself dynamically in convergence to non-hyperbolic fixed points under energy minimizing flows. In such a situation one has, quite generally, a center manifold such that coordinates on it describe degrees of freedom which do not decay exponentially fast. In appropriate settings (e.g. quiver representations) one can describe the dynamics on the center manifold combinatorially in terms of modular lattices. On certain walls in the parameter space iterated logarithms appear in the description of the asymptotics. As an application one gets a canonical weight-type filtration (labelled by reals) on any finite-dimensional representation. Conjecturally, this filtration describes asymptotics of certain geometric heat-type and Laplace-type PDEs. This is a joint project with Katzarkov, Kontsevich, and Pandit.

A quasi-Coxeter category is a braided tensor category which carries an action of a generalised braid group B_W on the tensor powers of its objects. The data which defines the action of B_W is similar in flavour to the associativity constraints in a monoidal category, but is related to the coherence of a family of fiber functors on C. I will outline how to construct such a structure on integrable, category O representations of a symmetrisable Kac-Moody algebra g, in a way that incorporates the monodromy of the KZ and Casimir connections of g. The rigidity of this structure implies in particular that the monodromy of the latter connection is given by the quantum Weyl group operators of the quantum group U_h(g). This is joint work with Andrea Appel.

The talk is based on the joint work with Lev Rozansky. I will overview our joint work where provide a catogification of HOMFLY-PT polynomial. The construction provide an interpretation of the knot homology as homology of coherent sheaves on the nested Hilbert scheme of points on the plane. In the second part of my talk I show how one can compute these homology explicitly for the large class of knots that includes in particular torus knots T_{n,1+kn}.

Abstract: I will explain recent work with Aaron Pixton and Junliang Shen which yields a proof of the Igusa cusp form conjecture. The conjecture says that the Donaldson-Thomas invariants of the product of a K3 surface and an elliptic curve are the coefficients of a Siegel modular form, the reciprocal of the Igusa cusp form. The proof uses a combination of sheaf theoretic methods (in particular derived autoequivalences) (with J. Shen) and new results in the Gromov-Witten theory of elliptic fibrations (with A. Pixton).

Abstract: The goal of my talk (based on joint work with Dima Grigoriev, Anatol Kirillov, and Gleb Koshevoy) is to generalize the celebrated Robinson-Schensted-Knuth (RSK) bijection between the set of matrices with nonnegative integer entries, and the set of the planar partitions.

Namely, for any pair of injective valuations on an integral domain we construct a canonical bijection K, which we call the generalized RSK, between the images of the valuations, i.e., between certain ordered abelian monoids.

Given a semisimple or Kac-Moody group, for each reduced word ii=(i_1,...,i_m) for a Weyl group element we produce a pair of injective valuations on C[x_1,...,x_m] and argue that the corresponding bijection K=K_ii, which maps the lattice points of the positive octant onto the lattice points of a convex polyhedral cone in R^m, is the most natural generalization of the classical RSK and, moreover, K_ii can be viewed as a bijection between Lusztig and Kashiwara parametrizations of the dual canonical basis in the corresponding quantum Schubert cell.

Generalized RSKs are abundant in "nature", for instance, any pair of polynomial maps phi,psi:C^m-->C^m with dense images determines a pair of injective valuations on C[x_1,...,x_n] and thus defines a generalized RSK bijection K_{phi,psi} between two sub-monoids of Z_+^m.

When phi and psi are birational isomorphisms, we expect that K_{phi,psi} has a geometric ``mirror image", i.e., that there is a rational function f on C^m whose poles complement the image of phi and psi so that the tropicalization of the composition psi^{-1}phi along f equals to K_{phi,psi}. We refer to such a geometric data as a (generalized) geometric RSK, and view f as a "super-potential." This fully applies to each ii-RSK situation, and we find a super-potential f=f_ii which helps to compute K_ii.

While each K_ii has a "crystal" flavor, its geometric (and mirror) counterpart f_ii emerges from the cluster twist of the relevant double Bruhat cell studied by Andrei Zelevinsky, David Kazhdan, and myself.

Abstract: We relate certain irregular Riemann-Hilbert maps to Gelfand-Zeitlin systems. As a quantum analogue, we show that the (quantum) Stokes factors of a Knizhnik-Zamolodchikov type equation satisfy Yang-Baxter equation. This part is joint with V. Toledano Laredo. We then discuss some related questions.

Abstract: Okounkov and Pandharipande proved that Gromov-Witten invariants of an elliptic curve are coefficients of quasimodular forms. I will explain how to lift this quasimodularity to the level of cycles (on the moduli space of stable curves). This is joint work with Georg Oberdieck.

Motivated by mirror symmetry, we study the counting of open curves in log Calabi-Yau surfaces. Although we start with a complex surface, the counting is achieved by applying methods from Berkovich geometry (non-archimedean analytic geometry). This gives rise to new geometric invariants inaccessible by classical methods. These invariants satisfy a list of very nice properties and can be computed explicitly. If time permits, I will mention the conjectural wall-crossing formula, relations with the works of Gross-Hacking-Keel and applications towards mirror symmetry.

Representation theory studies ways to represent algebraic objects (such as associative algebras) as algebras of linear operators on vector spaces. Physics has a long record of supplying interesting algebras to study. In my talk I will consider algebras that arise as quantizations of moduli spaces of instantons. I will review known results about representations of these algebras, mention their relevance to Physics, and, time permitting, outline some prospective developments: representations over fields of positive characteric are coming!

For a k-tuple of conjugacy classes in GL_n(C) we consider the corresponding genus 0 character variety, i.e. the moduli space of local systems on P^1 minus k punctures with prescribed local monodromies at the punctures. In the case when at least one of the conjugacy classes is regular semisimple and the eigenvalues are generic we construct a stratification of the character variety in such a way that each stratum is a product of symplectic torus of dimension 2(d-i) and affine space of dimension i, where 2d is the dimension of the variety. I will explain how this implies the so-called curious hard Lefschetz conjecture of Hausel, Letellier and Villegas, and how their conjecture on mixed Hodge polynomials of character varieties is related to the conjectures of Gorsky, Negut, Oblomkov, Rasmussen and Shende, which connect Khovanov-Rozansky homology of links with K-theory of the Hilbert scheme.

I will discuss joint work with Zsuzsanna Dancso and Vivek Shende on a variant of the P=W conjecture. Let C be a smooth Riemann surface. de Cataldo, Hausel, and Migliorini conjecture that the perverse filtration on the cohomology of the moduli of higgs bundles on C equals the weight filtration on the character variety of C, under the non-abelian hodge correspondence.

We define and prove an analogous conjecture in rank one, where C is replaced by a nodal collection of rational curves.

Given a curve X of genus g, the moduli stack of Higgs sheaves of rank r and degree d has dimension 2(g-1)r^2. It can be viewed as the cotangent stack of the stack of coherent sheaves of type (r,d) over X, and Laumon proved that the substack of nilpotent Higgs pairs is Lagrangian. This substack is a global analog of the nilpotent cone, and is nothing but the 0-fiber of the Hitchin map. It is highly singular, and one first interesting step toward its comprehension is the study of its irreducible components. This study is also motivated by a conjecture stating that the number of stable components is given by the value at 1 of the Kac polynomial of the quiver with one vertex and g loops (Hausel, Letellier, Rodriguez Villegas), as well as by the W=P conjecture (de Cataldo, Hausel, Migliorini). I will give a nice combinatorial description of this set of components, and will explain which ones subsist when we restrict ourselves to the stable locus (with respect to the usual slope stability).

The quantum connection for the Springer resolution is identified with Cherednik's affine Knizhnik-Zamolodchikov connection by Braverman, Maulik and Okounkov. For a parabolic subgroup P, I will talk about how to compute the quantum connection of T^*(G/P) using stable basis constructed by Maulik and OKounkov, confirming a conjecture of Braverman.

Recently, Templier and Lam proved the mirror conjecture for minuscule G/P by identifying the quantum connection, Frenkel-Gross connection and Kloosterman sheaves. I will construct a regular connection on P^1\{0,1,\infty}, an identify it with the quantum connection for T^*(G/P) when P is minuscule.

One particularly important class of integrable systems are the Painleve/Garnier/Schlesinger-type systems that arise as "monodromy-preserving" flows in families of linear differential equations. It turns out that one can translate these flows into essentially geometric terms, in which the equations are sheaves on a suitable surface, and the (discrete) flows are twisting by a line bundle, with one caveat: the surface involved is noncommutative. I'll explain how this works, and sketch a number of consequences, e.g., how derived equivalences between noncommutative elliptic surfaces give rise to new isomonodromic descriptions for the Painleve equations.

The moduli spaces of local systems on marked surfaces enjoy many nice properties. In particular, it was shown by Fock and Goncharov that they form examples of cluster varieties, which means that they are Poisson varieties with a positive atlas of toric charts, and thus admit canonical quantizations. I will describe joint work with A. Shapiro in which we embed the quantized enveloping algebra U_q(sl_n) into the quantum character variety associated to a punctured disk with two marked points on its boundary. The construction is closely related to the (quantized) multiplicative Grothendieck-Springer resolution for SL_n. I will also explain how the R-matrix of U_q(sl_n) arises naturally in this topological setup as a (half) Dehn twist. Time permitting, I will describe some potential applications to the study of positive representations of the split real quantum group U_q(sl_n,R).

In this talk I will discuss some aspects of representation theory of quantizations of Nakajima quiver varieties. These quantizations are algebras depending on parameters. I will introduce wall-crossing functors relating algebras with different parameters. Then I will discuss structural results about these functors and their applications (partly joint with Bezrukavnikov).

Affine W-algebras are a somewhat complicated family of (topological) associative algebras associated with a semisimple Lie algebra, quantizing functions on the algebraic loop space of Kostant's slice. They have attracted a great deal of attention because of Feigin-Frenkel's duality theorem for them, which identifies W-algebras for a Lie algebra and for its Langlands dual through a subtle construction.

The purpose of this talk is threefold: 1) to introduce a "stratification" of the category of modules for the affine W-algebra, 2) to prove an analogue of Skryabin's equivalence in this setting, realizing the category of (discrete) modules over the W-algebra in a more natural way, and 3) to explain how these constructions help understand Whittaker categories in the more general setting of local geometric Langlands. These three points all rest on the same geometric observation, which provides a family of affine analogues of Bezrukavnikov-Braverman-Mirkovic. These results lead to a new understanding of the exactness properties of the quantum Drinfeld-Sokolov functor.

The usual AGT-W relations connect U(r) gauge theory with the vertex operator algebra W(r). In this talk, I will prove a q-deformation of this result, by using the action of the double shuffle algebra, also known as W(infinity), on its level r representation.

I will present a recent conjecture that connects the geometry of compactified Jacobians of unibranch plane singularities with the DAHA-superpolynomials of algebraic knots. This somewhat resembles the famous Kazhdan-Lusztig conjecture, but (double) Hecke algebras describe here a different kind of geometry, directly related to that of p-adic orbital integrals (in Fundamental Lemma) and theory of affine Springer fibers (the anisotropic case, type A).

The key geometric ingredients of our construction are degrees and dimensions of the generalized Lusztig-Smelt-Piontkowski cells of the flagged Jacobian factors (new objects, to be defined from scratch); there is a connection with the Kazhdan-Lusztig dimension formulas from their 1988 paper, proved later by Bezrukavnikov.

The DAHA-superpolynomials are expected to coincide with the Khovanov-Rozansky polynomials of algebraic knots. They depend on the parameters a,q,t; for instance, a=-1,q=t is the case of the Alexander polynomials, which can be directly expressed via the corresponding singularities (no Jacobian factors are necessary).

When a=0,q=1,t=1/p, we conjecture them to coincide with the p-adic orbital integrals; understanding any "a,q" in the theory of these integrals is a challenge. Our conjecture readily implies that the orbital integrals in type A (!) depend only on the topological (not just analytic) type of singularity, which is another challenge.

Particular instances of our conjecture are due to Gorsky-Mazin (a=0, torus knots); the so-called Cherednik-Danilenko conjecture (any algebraic knots for a=0,q=1). Our conjecture is different from that due to Oblomkov-Rasmussen-Schende; if time permits I will provide some details here via a reformulation of our conjecture in terms of the weight filtration.

This is joint with Ivan Danilenko and Ian Philipp. I may skip some details concerning the DAHA construction, but the geometric superpolynomials will be defined in full (from scratch). Our conjecture is checked by now "beyond a reasonable doubt", incl. quite involved examples when the Piontkowski cells are not affine.

The first part (at the Lie Groups Seminar) will be mostly about the plane curve singularities and the corresponding geometric superpolynomials. I will briefly discuss the DAHA construction and the relations to affine Springer fibers; plane curve singularities and Jacobian factors will be fully defined (this is an entirely local theory, where the compactified Jacobians are not too involved).

Then (at "Geometric Representation Theory") we will provide details concerning the DAHA construction (any root systems and iterated knots); this is in fact a one-line formula (not much from DAHA theory is really needed). It time permits, its topological invariance will be justified and further relations to orbital integrals and topology will be discussed including the ORS paper and that by them with Gorsky.

Jones polynomials and WRT invariants are well-known invariants of links in S^3. Their categorification attracts a lot of attention now. The key numerical invariant here is the Poincare polynomial of the triply graded Khovanov-Rozansky homology, also called HOMFLYPT homology. In spite of recent developments, this theory remains very difficult apart from the celebrated Khovanov homology (the case of sl(2)) with very few known formulas (only for the simplest uncolored knots). Several alternative approaches to these polynomials were suggested recently (the connections are mostly conjectural). We will discuss the direction based on DAHA, which was recently extended from torus knots to arbitrary torus iterated links (including all algebraic links). The talk will be mostly focused on the DAHA-Jones polynomial of type A_1, which is essentially sufficient to understand the procedures we employ and to see which steps are needed to go from torus knots to arbitrary iterated knots/links. Based on our joint works with Ivan Cherednik.

Cluster algebras are subalgebras of the field of rational functions in n variables, whose generators are given by recursive application of cluster mutation to the standard monomials. Quantum cluster algebras are a certain very natural quantum deformation of these algebras. Although the definition of cluster mutation is entirely elementary, the question of proving the positivity of the coefficients that appear after repeated cluster mutation has remained open. I'll explain a proof, via a proof of a conjecture of Kontsevich and Efimov regarding the Hodge theory of moduli spaces appearing in cohomological Donaldson-Thomas theory.

The Gopakumar-Vafa invariants are integer valued invariants on Calabi-Yau 3-folds which are expected to be related to other curve counting invariants on them such as Gromov-Witten invariants, Pandharipande-Thomas invariants. However a mathematically rigorous definition of GV invariants is not obvious. In 2012, Kiem-Li used perverse sheaves of vanishing cycles to define GV invariants, which modify the previous definition by Hosono-Saito-Takahashi. In this talk, I will explain that Kiem-Li's definition is not deformation invariant, so not a correct one. I will modify the definition of Kiem-Li, by introducing the notion of Calabi-Yau d-critical structures. The main result is that our definition of GV invariants with irreducible curve classes match with PT invariants for local surfaces. This is a joint work with Davesh Maulik.

The Jucys-Murphy elements are known to generate a maximal commutative subalgebra in the Hecke algebra. They can be categorified to a family of commuting complexes of Soergel bimodules. I will describe a relation between a category generated by these complexes and the category of sheaves on the flag Hilbert scheme of points on the plane, using the recent work of Elias and Hogancamp on categorical diagonalization. As an application, I will give an explicit conjectural description of the Khovanov-Rozansky homology of generalized torus links. The talk is based on a joint work with Andrei Neguț and Jacob Rasmussen.