Pramod Achar, Louisiana State University
Conjectures on p-cells, tilting modules, and nilpotent orbits
For quantum groups at a root of unity, there is a web of theorems (due to Bezrukavnikov and Ostrik, and relying on work of Lusztig) connecting the following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits; and (iii) Kazhdan–Lusztig cells in the affine Weyl group. In this talk, I will review these results, and I will explain a (partly conjectural) analogous picture for reductive algebraic groups over fields of positive characteristic, inspired by a conjecture of Humphreys. This is joint work with W. Hardesty and S. Riche.
Ana Balibanu, Harvard University
Perverse sheaves and the cohomology of regular Hessenberg varieties
Hessenberg varieties are a distinguished family of projective varieties associated to a semisimple complex algebraic group. We use the formalism of perverse sheaves to study their cohomology rings. We give a partial characterization, in terms of the Springer correspondence, of the irreducible representations which appear in the action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. We also prove a support theorem for the universal family of regular Hessenberg varieties, and we deduce that its fibers, though not necessarily smooth, always have the "Kahler package". This is joint work with Peter Crooks.
Roman Bezrukavnikov, Massachusetts Institute of Technology
Modular representations and perverse sheaves on affine flag varieties
I will give an overview of a joint project with Simon Riche and Laura Rider and another one with Dima Arinkin aimed at a modular version of the equivalence between two geometric realization of the affine Hecke algebra and derived Satake equivalence respectively. As a byproduct we obtain a proof of the Finkelberg-Mirkovic conjecture and a possible approach to understanding cohomology of higher Frobenius kernels with coefficients in a G-module.
Tristan Bozec, University of Montpellier
Relative critical loci, quiver moduli, and new lagrangian subvarieties
The preprojective algebra of a quiver naturally appears when computing the cotangent to the quiver moduli, via the moment map. When considering the derived setting, it is replaced by its differential graded (dg) variant, introduced by Ginzburg. This construction can be generalized using potentials, so that one retrieves critical loci when considering moduli of perfect modules. Our idea is to consider some relative, or constrained critical loci, deformations of the above, and study Calabi--Yau structures on the underlying relative versions of Ginzburg's dg algebras. It yields for instance some new lagrangian subvarieties of the Hilbert schemes of points on the plane.
This reports a joint work with Damien Calaque and Sarah Scherotzke
arxiv.org/abs/2006.01069
Justin Campbell, California Institute of Technology
Geometric class field theory and Cartier duality
I will explain a generalized Albanese property for smooth curves, which implies Deligne's geometric class field theory with arbitrary ramification. The proof essentially reduces to some well-known Cartier duality statements. This is joint work with Andreas Hayash.
Gurbir Dhillon, Stanford University
Fundamental local equivalences in quantum geometric Langlands
In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory--Lurie proposed a conjectural substitute, later termed the fundamental local equivalence, relating categories of arc-integrable Kac--Moody representations and Whittaker D-modules on the affine Grassmannian. With a few exceptions, we verified this conjecture non-factorizably, as well as its extension to the affine flag variety. This is a report on joint work with Justin Campbell and Sam Raskin.
Jens Eberhardt, Max Planck Institute, Bonn
K-Motives and Koszul Duality
Koszul duality, as conceived by Beilinson-Ginzburg-Soergel, describes a remarkable symmetry in the representation theory of Langlands dual reductive groups. Geometrically, Koszul duality can be stated as an equivalence of categories of mixed (motivic) sheaves on flag varieties. In this talk, I will argue that there should be an an 'ungraded' version of Koszul duality between monodromic constructible sheaves and equivariant K-motives on flag varieties. For this, I will explain what K-motives are and present preliminary results.
Ben Elias, University of Oregon
Categorification of the Hecke algebra at roots of unity.
Categorical representation theory is filled with graded additive categories (defined by generators and relations) whose Grothendieck groups are algebras over \mathbb{Z}[q,q^{-1}]. For example, Khovanov-Lauda-Rouquier (KLR) algebras categorify the quantum group, and the diagrammatic Hecke categories categorify Hecke algebras. Khovanov introduced Hopfological algebra in 2006 as a method to potentially categorify the specialization of these \mathbb{Z}[q,q^{-1}]-algebras at q = \zeta_n a root of unity. The schtick is this: one equips the category (e.g. the KLR algebra) with a derivation d of degree 2, which satisfies d^p = 0 after specialization to characteristic p, making this specialization into a p-dg algebra. The p-dg Grothendieck group of a p-dg algebra is automatically a module over \mathbb{Z}[\zeta_{2p}]... but it is NOT automatically the specialization of the ordinary Grothendieck group at a root of unity!
Upgrading the categorification to a p-dg algebra was done for quantum groups by Qi-Khovanov and Qi-Elias. Recently, Qi-Elias accomplished the task for the diagrammatic Hecke algebra in type A, and ruled out the possibility for most other types. Now the question is: what IS the p-dg Grothendieck group? Do you get the quantum group/hecke algebra at a root of unity, or not?
This is a really hard question, and currently the only techniques for establishing such a result involve explicit knowledge of all the important idempotents in the category. These techniques sufficed for quantum \mathfrak{sl}_n with n \le 3, but new techniques are required to make further progress.
After reviewing the theory of p-dg algebras and their Grothendieck groups, we will present some new techniques and conjectures, which we hope will blow your mind.
Everything is joint with You Qi.
Michael Finkelberg, National Research University Higher School of Economics
Global Demazure modules
The Beilinson-Drinfeld Grassmannian of a simple complex algebraic group admits a natural stratification into "global spherical Schubert varieties". In the case when the underlying curve is the affine line, we determine algebraically the global sections of the determinant line bundle over these global Schubert varieties as modules over the corresponding Lie algebra of currents. The resulting modules are the global Weyl modules (in the simply laced case) and generalizations thereof. This is a joint work with Ilya Dumanski and Evgeny Feigin.
Eugene Gorsky, University of California, Davis
Parabolic Hilbert schemes via the Dunkl-Opdam subalgebra
In this note we give an alternative presentation of the rational Cherednik algebra H_c corresponding to the permutation representation of S_n. As an application, we give an explicit combinatorial basis for all standard and simple modules if the denominator of c is at least n, and describe the action of H_c in this basis. We also give a basis for the irreducible quotient of the polynomial representation and compare it to the basis of fixed points in the homology of the parabolic Hilbert scheme of points on the plane curve singularity {x^n=y^m}. This is a joint work with José Simental and Monica Vazirani.
Oscar Kivinen, California Institute of Technology
Z-algebras from Coulomb branches
I will explain how to obtain the Gordon-Stafford construction and some related constructions of Z-algebras in the literature, using certain mathematical avatars of line defects in 3d N=4 theories. Time permitting, I will discuss the K-theoretic and elliptic cases as well.
Martina Lanini, Università di Roma Tor Vergata
Singularities of Schubert varieties within a right cell
We describe an algorithm which takes as input any pair of permutations and gives as output two permutations lying in the same Kazhdan-Lusztig right cell. There is an isomorphism between the Richardson varieties corresponding to the two pairs of permutations which preserves the singularity type. This fact has applications in the study of W-graphs for symmetric groups, as well as in finding examples of reducible associated varieties of sln-highest weight modules, and comparing various bases of irreducible representations of the symmetric group or its Hecke algebra. This is joint work with Peter McNamara.
Michael McBreen, Aarhus University & Harvard University
Elliptic stable envelopes via loop spaces
Elliptic stable envelopes, introduced by Aganagic and Okounkov, are a key ingredient in the study of quantum integrable systems attached to a symplectic resolution. I will describe a relation between elliptic stable envelopes on a hypertoric variety and a certain 'loop space' of that variety. Joint with Artan Sheshmani and Shing-Tung Yau.
Tudor Padurariu, Massachusetts Institute of Technology
K-theoretic Hall algebras for quivers with potential
Given a quiver with potential, Kontsevich-Soibelman constructed a Hall algebra on the cohomology of the stack of representations of (Q,W). In particular cases, one recovers positive parts of Yangians as defined by Maulik-Okounkov. For general (Q,W), the Hall algebra has nice structure properties, for example Davison-Meinhardt proved a PBW theorem for it using the decomposition theorem.
One can define a K-theoretic version of this algebra using certain categories of singularities that depend on the stack of representations of (Q,W). In particular cases, these Hall algebras are positive parts of quantum affine algebras. We show that some of the structure properties in cohomology, such as the PBW theorem, can be lifted to K-theory, replacing the use of the decomposition theorem with semi-orthogonal decompositions.
Tomasz Przezdziecki, The University of Edinburgh
An extension of Suzuki's functor to the critical level
Suzuki's functor relates the representation theory of the affine Lie algebra to the representation theory of the rational Cherednik algebra in type A. In this talk, we discuss an extension of this functor to the critical level, t=0 case. This case is special because the respective categories of representations have large centres. Our main result describes the relationship between these centres, and provides a partial geometric interpretation in terms of Calogero-Moser spaces and opers.
Jenna Rajchgot, University of Saskatchewan
Type D quiver representation varieties, double Grassmannians, and symmetric varieties
Since the 1980s, mathematicians have found connections between orbit closures in type A quiver representation varieties and Schubert varieties in type A flag varieties. For example, singularity types appearing in type A quiver orbit closures coincide with those appearing in Schubert varieties in type A flag varieties (Bobinski-Zwara); combinatorics of type A quiver orbit closure containment is governed by Bruhat order on the symmetric group (follows from work of Zelevinsky, Kinser-R); and multiple researchers have produced formulas for classes of type A quiver orbit closures in equivariant cohomology and K-theory in terms of Schubert polynomials, Grothendieck polynomials, and related objects.
After recalling some of this type A story, I will discuss joint work with Ryan Kinser on type D quiver representation varieties. I will describe explicit embeddings which completes a circle of links between orbit closures in type D quiver representation varieties, B-orbit closures (for a Borel subgroup B of GL_n) in certain symmetric varieties GL_n/K, and B-orbit closures in double Grassmannians Gr(a, n) x Gr(b, n). I will end with some geometric and combinatorial consequences, as well as a brief discussion of joint work in progress with Zachary Hamaker and Ryan Kinser on formulas for classes of type D quiver orbit closures in equivariant cohomology.
Sam Raskin, University of Texas at Austin
Tate's thesis in the de Rham setting
This is joint work with Justin Hilburn. We will explain a theorem showing that D-modules on the Tate vector space of Laurent series are equivalent to ind-coherent sheaves on the space of rank 1 de Rham local systems on the punctured disc equipped with a flat section. Time permitting, we will also describe an application of this result in the global setting. Our results may be understood as a geometric refinement of Tate's ideas in the setting of harmonic analysis. They also may be understood as a proof of a strong form of the 3d mirror symmetry conjectures: our results amount to an equivalence of A/B-twists of the free hypermultiplet and a U(1)-gauged hypermultiplet.
Laura Rider, University of Georgia
Centralizer of a regular unipotent element and perverse sheaves on the affine flag variety
In this talk, I will give a geometric description of the category of representations of the centralizer of a regular unipotent element in a reductive algebraic group in terms of perverse sheaves on the Langlands dual affine flag variety. This is joint work with R. Bezrukavnikov and S. Riche.
Anna Romanov, University of Sydney
A categorification of the Lusztig—Vogan module
Admissible representations of real reductive Lie groups are a key player in the world of unitary representation theory. The characters of irreducible admissible representations were described by Lustig—Vogan in the 80’s in terms of a geometrically-defined module over the associated Hecke algebra. In this talk, I’ll describe a categorification of this module using Soergel bimodules, with a focus on examples. This is work in progress.
Pavel Safronov, University of Zurich
Parabolic restriction for Harish-Chandra bimodules and dynamical R-matrices
The category of Harish-Chandra bimodules is ubiquitous in representation theory. In this talk I will explain their relationship to the theory of dynamical R-matrices (going back to the works of Donin and Mudrov) and quantum moment maps. I will also relate the monoidal properties of the parabolic restriction functor for Harish-Chandra bimodules to the so-called standard dynamical R-matrix. This is a report on work in progress, joint with Artem Kalmykov.
Sarah Scherotzke, University of Luxembourg
Cotangent complexes of moduli spaces and Ginzburg dg algebras
We give an introduction to the notion of moduli stack of a dg category. We explain what shifted symplectic structures are and how they are connected to Calabi-Yau structures on dg categories. More concretely, we will show that the cotangent complex to the moduli stack of a dg category A admits a modular interpretation: namely, it is isomorphic to the moduli stack of the *Calabi-Yau completion* of A. This answers a conjecture of Keller-Yeung. The talk is based on joint work
This is joint work with Damien Calaque and Tristan Bozec
arxiv.org/abs/2006.01069
Olivier Schiffmann, CNRS, Université de Paris-Sud Orsay
Yangians and cohomological Hall algebras of Higgs sheaves on curves
We will review a set of conjectures related to the structure of cohomological Hall algebras (COHA) of categories of Higgs sheaves on curves. We then focus on the case of P^1, and relate its COHA to the affine Yangian of sl_2.
Monica Vazirani, University of California, Davis
The ``Springer" representation of the DAHA
The Springer resolution and resulting Springer sheaf are key players in geometric representation theory. While one can construct the Springer sheaf geometrically, Hotta and Kashiwara gave it a purely algebraic reincarnation in the language of equivariant $D(\mathfrak{g})$-modules. For $G = GL_N$, the endomorphism algebra of the Springer sheaf, or equivalently of the associated $D$-module, is isomorphic to $\mathbb{C}[\mathcal{S}_n]$ the group algebra of the symmetric group. In this talk, I'll discuss a quantum analogue of this.
In joint work with Sam Gunningham and David Jordan, we define quantum Hotta-Kashiwara $D$-modules $\mathrm{HK}_\chi$, and compute their endomorphism algebras. In particular $\mathrm{End}_{\mathcal{D}_q(G)}(\mathrm{HK}_0) \simeq \mathbb{C}[\mathcal{S}_n]$. This is part of a larger program to understand the category of strongly equivariant quantum $D$-modules.
Our main tool to study this category is Jordan's elliptic Schur-Weyl duality functor to representations of the double affine Hecke algebra (DAHA). When we input $\mathrm{HK}_0$ into Jordan's functor, the endomorphism algebra over the DAHA of the output is $\mathbb{C}[\mathcal{S}_n]$ from which we deduce the result above.
From studying the output of all the $\mathrm{HK}_\chi$, we are able to compute that for input a distinguished projective generator of the category the output is the DAHA module generated by the sign idempotent.
This is joint work with Sam Gunningham and David Jordan.
