# Geometric Representation Theory

COVID-19 information for PI Residents and Visitors

Conference Date:
Monday, June 22, 2020 (All day) to Friday, June 26, 2020 (All day)
Scientific Areas:
Mathematical Physics

Update!
The talks will be broadcast over Zoom. This will be via an institutional account (through either PI or the MPI). In order to participate please register by using the link to the registration form provided below. More information will follow.

There will be an opportunity for participants to contribute a 5-minute pre-recorded talk. If you are interested in participating, please enter the details in the registration form by May 31, 2020.

Originally we had planned to have a twinned conference held simultaneously at Max Planck Institute in Bonn, Germany and Perimeter Institute in Waterloo, Canada. In light of the current Covid-19 pandemic, we have decided that the conference "Geometric Representation Theory" will instead go forward as an online event. The concept of the twinned conference was motivated by the desire to reduce environmental impact of conference travels. Our hope is that this initiative will help reduce transatlantic flights, while still promoting long distance interactions.

In recent years, tools from algebraic geometry and mathematical physics have proven very influential in representation theory.  The most famous example is the geometric Langlands program, which is inspired by the original Langlands program in the function field case, but has also included advances in the theory of algebraic groups, Lie algebras and Cherednik algebras.  The goal of this twinned conference is to bring together experts in geometric representation theory and adjacent areas to discuss the forefront of current developments in this highly active field.

Registration via the North American event is now closed.
To register via the Max Planck website, please clck here

• Pramod Achar, Louisiana State University
• Ana Balibanu, Harvard University
• Roman Bezrukavnikov, Massachusetts Institute of Technology
• Tristan Bozec, University of Montpellier
• Justin Campbell, California Institute of Technology
• Gurbir Dhillon, Stanford University
• Jens Eberhardt, Max Planck Institute, Bonn
• Ben Elias, University of Oregon
• Michael Finkelberg, National Research University Higher School of Economics
• Eugene Gorsky, University of California, Davis
• Oscar Kivinen, California Institute of Technology
• Martina Lanini, Università di Roma Tor Vergata
• Michael McBreen, Aarhus University & Harvard University
• Tudor Padurariu, Massachusetts Institute of Technology
• Tomasz Przezdziecki, The University of Edinburgh
• Jenna Rajchgot, University of Saskatchewan
• Sam Raskin, University of Texas at Austin
• Laura Rider, University of Georgia
• Anna Romanov, University of Sydney
• Pavel Safronov, University of Zurich
• Sarah Scherotzke, University of Luxembourg
• Olivier Schiffmann, CNRS, Université de Paris-Sud Orsay
• Monica Vazirani, University of California, Davis
• Salih Abdalla, National Industrial Training Institute
• Khalid Ibrahim Adam Ahmed, Najran University
• Aswin Balasubramanian, Rutgers University
• Gabriela Barcenas, Universidad de Guanajuato
• Marc Besson, University of North Carolina Chapel Hill
• Elijah Bodish, University of Oregon
• Harrison Chen, Cornell University
• Zhe Chen, Shantou University
• Jingren Chi, University of Maryland College Park
• Nico Courts, University of Washington
• Peter Crooks, Northeastern University
• Daniel Chupin, University of California, Berekeley
• Kaustav Das, Indian Institute of Engineering Science and Technology
• Roukaya Dekhil, Ludwig Maximilian University of Munich
• Tanmay Deshpande, Tata Institute of Fundamental Research Mumbai
• Ruifeng Dong, State University of New York at Buffalo
• Anne Dranowski, University of Toronto
• Drew Duffield, Durham University
• Dotsenko Egor, Higher School of Economics
• Tom Gannon, University of Texas at Austin
• Nicolle Gonzalez, University of California Los Angeles
• Pallav Goyal, University of Chicago
• Nikolay Grantcharov, University of Chicago
• Zhengping Gui, Tsinghua University
• Meng Guo, Perimeter Institute
• Iva Halacheva, Northeastern University
• Andrew Hardt, University of Minnesota
• Jiuzu Hong, University of North Carolina at Chapel Hill
• You Hung Hsu, National Center for Theoretical Sciences
• Mengwei Hu, Sichuan University
• Mee Seong Im, United States Military Academy
• Nafiz Ishtiaque, Institute for Advanced Study
• Ankur Jyoti Kalita, Gauhati University
• Tina Kanstrup, University of Massachusetts
• Banani Kashyap, University of Delhi
• Hannah Keese, Cornell University
• Hyungseop Kim, University of Toronto
• Ryosuke Kodera, Kobe University
• Hitoshi Konno, Tokyo University of Marine Science and Technology
• Ethan Kowalenko, University of California Riverside
• Nicholas Lai, University of British Columbia
• Patrick Lank, University of New Mexico
• Ian Le, Northwestern University
• Aolong Li, Indiana University Bloomington
• Cailan Li, Columbia University
• Jianrong Li, University of Graz
• Henry Liu, Columbia University
• Alvaro Martinez, Columbia University
• Matthew McMillan, University of California, Los Angeles
• Benedict Morrissey, University of Pennsylvania
• Calder Morton-Ferguson, Massachusetts Institute of Technology
• Kaveh Mousavand, Université du Québec à Montréal
• Sanat Mulay, University of Southern California
• Hiroshi Naruse, University of Yamanashi
• Satoshi Nawata, Fudan University
• Gyujin Oh, Princeton University
• Md Abdur Rahman, Ryerson University
• Anna Romanov, University of Sydney
• Daniele Rosso, Indiana University Northwest
• Lev Rozansky, University of North Carolina
• Arghya Sadhukhan, University of Maryland
• Guillermo Sanmarco, Universidad Nacional de Córdoba
• Peng Shan, Tsinghua University
• Eric Sharpe, Virginia Polytechnic Institute and State University
• Gicheol Shin, Korea National University of Education
• Pavel Shlykov, University of Toronto
• Jose Simental Rodriguez, University of California, Davis
• Changjian Su, University of Toronto
• Aiden Suter, Perimeter Institute
• Jackson Van Dyke, University of Texas at Austin
• Jonathan Wang, Massachusetts Institute of Technology
• Alex Weekes, University of British Columbia
• Linfeng Wei, Sichuan University
• Dwight Williams II, University of Texas at Arlington
• Yaping Yang, University of Melbourne
• Victor Zhang, California Institute of Technology
• Shan Zhou, University of California Santa Barbara

Monday, June 22, 2020

 Time Event Location 10:15 – 10:45 EST Ben Webster, Perimeter Institute & University of WaterlooWelcome and Opening Remarks Virtual 10:45 – 11:55 EST Martina Lanini, Università di Roma Tor VergataSingularities of Schubert varieties within a right cell Virtual 12:00 – 12:55 EST Olivier Schiffmann, CNRS, Université de Paris-Sud OrsayYangians and cohomological Hall algebras of Higgs sheaves on curves Virtual 14:00 – 14:55 EST Samuel Raskin, University of Texas, AustinTate's thesis in the de Rham setting Virtual 15:15 – 15:45 EST Gurbir Dhillion, Stanford UniversityFundamental local equivalences in quantum geometric Langlands Virtual

Tuesday, June 23, 2020

 Time Event Location 10:00 – 10:30 EST Oscar Kivinen, California Institute of TechnologyZ-algebras from Coulomb branches Virtual 10:45 – 11:55 EST Sarah Scherotzke, University of LuxembourgCotangent complexes of moduli spaces and Ginzburg dg algebras Virtual 12:00 – 12:55 EST Laura Rider, University of GeorgiaCentralizer of a regular unipotent element and perverse sheaves on the affine flag variety Virtual 14:00 – 14:55 EST Jenna Rajchgot, University of SaskatchewanType D quiver representation varieties, double Grassmannians, and symmetric varieties Virtual 15:15 – 15:45 EST Tudor Padurariu, Massachusetts Institute of TechnologyK-theoretic Hall algebras for quivers with potential Virtual

Wednesday, June 24, 2020

 Time Event Location 10:00 – 10:30 EST Michael McBreen, Aarhus University & Harvard UniversityElliptic stable envelopes via loop spaces Virtual 10:45 – 11:55 EST Michael Finkelberg, National Research University Higher School of EconomicsGlobal Demazure modules Virtual 12:00 – 12:55 EST Roman Bezrukavnikov, Massachusetts Institute of TechnologyModular representations and perverse sheaves on affine flag varieties Virtual 14:00 – 14:55 EST Monica Vazirani, University of California, DavisThe Springer" representation of  the DAHA Virtual 15:15 – 15:45 EST Justin Campbell, California Institute of TechnologyGeometric class field theory and Cartier duality Virtual

Thursday, June 25, 2020

 Time Event Location 11:15 – 11:45 EST Tristan Bozec, University of MontpellierRelative critical loci, quiver moduli, and new lagrangian subvarieties Virtual 12:00 – 12:55 EST Pavel Safronov, University of ZurichParabolic restriction for Harish-Chandra bimodules and dynamical R-matrices Virtual 14:00 – 14:55 EST Eugene Gorsky, University of California, DavisParabolic Hilbert schemes via the Dunkl-Opdam subalgebra Virtual 15:15 – 15:45 EST Tomasz Przezdziecki, University of EdinburghAn extension of Suzuki's functor to the critical level Virtual 16:00 – 16:30 EST Anna Romanov, University of SydneyA categorification of the Lusztig—Vogan module Virtual

Friday, June 26, 2020

 Time Event Location 11:15 – 11:45 EST Jens Eberhardt, Max Planck Institute, BonnK-Motives and Koszul Duality Virtual 12:00 – 12:55 EST Pramod Achar, Louisiana State UniversityConjectures on p-cells, tilting modules, and nilpotent orbits Virtual 14:00 – 14:55 EST Ben Elias, University of OregonCategorification of the Hecke algebra at roots of unity. Virtual 15:15 – 15:45 EST Ana Balibanu, Harvard UniversityPerverse sheaves and the cohomology of regular Hessenberg varieties Virtual

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.

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.

Aswin Balasubramanian, Rutgers University

Presentation:  GRT_Recording_Balasubramanian.mp4

Poster:

Elijah Bodish, University of Oregon

Presentation:  GRT_Recording_Bodish.mp4

Poster:

Harrison Chen, Cornell University

Presentation:  GRT_Recording_Chen.mp4

Poster:

Peter Crooks, Northeastern University

Presentation:  GRT_Recording_Crooks.mp4

Poster:

Bertrand Nguefack, University of Yaounde

Presentation: ProjRepPreLieAgl2020.mp4

Poster:

Daniele Rosso, Indiana University Northwest

Presentation:

Poster:

Jose Simental Rodriguez, University of California, Davis

Presentation:  GRT_Recording_Simental.mp4

Poster:

## Perverse sheaves and the cohomology of regular Hessenberg varieties

Friday Jun 26, 2020
Speaker(s):

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.

Scientific Areas:

## Categorification of the Hecke algebra at roots of unity.

Friday Jun 26, 2020
Speaker(s):

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.

Scientific Areas:

## Conjectures on p-cells, tilting modules, and nilpotent orbits

Friday Jun 26, 2020
Speaker(s):

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.

Scientific Areas:

## K-Motives and Koszul Duality

Friday Jun 26, 2020
Speaker(s):

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.

Scientific Areas:

## A categorification of the Lusztig—Vogan module

Thursday Jun 25, 2020
Speaker(s):

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.

Scientific Areas:

## An extension of Suzuki's functor to the critical level

Thursday Jun 25, 2020
Speaker(s):

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.

Scientific Areas:

## Parabolic Hilbert schemes via the Dunkl-Opdam subalgebra

Thursday Jun 25, 2020
Speaker(s):

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

Scientific Areas:

## Parabolic restriction for Harish-Chandra bimodules and dynamical R-matrices

Thursday Jun 25, 2020
Speaker(s):

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.

Scientific Areas:

## Relative critical loci, quiver moduli, and new lagrangian subvarieties

Thursday Jun 25, 2020
Speaker(s):

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

Scientific Areas:

## Geometric class field theory and Cartier duality

Wednesday Jun 24, 2020

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.

Scientific Areas:

## Pages

Scientific Organizers:

• Tobias Barthel, Max Planck Institute, Bonn
• André Henriques, Oxford University
• Joel Kamnitzer, University of Toronto
• Carl Mautner, University of California, Riverside
• Aaron Mazel-Gee, University of Southern California
• Kevin Mcgerty, Oxford University
• Catharina Stroppel, Bonn University
• Ben Webster, Perimeter Institute & University of Waterloo