Schedule of the Workshop "Quantum geometric and algebraic representation theory"

Monday, October 16

10:15 - 10:50 Registration & Welcome coffee
10:50 - 11:00 Opening remarks
11:00 - 12:00 Wolfgang Soergel: Koszul duality for real groups
12:00 - 13:50 Lunch break
13:50 - 14:50 Jan Schröer: Geometric realization of crystal graphs and semicanonical bases
15:00 - 16:00 Yoshiki Oshima: The orbit method and characters of representations for real reductive groups
16:00 - 16:30 Tea and cake
16:30 - 17:30 Martina Lanini: Sheaves on the alcoves and modular representations
17:40 - 18:40 Tina Kanstrup: Braid group actions, matrix factorizations and link invariants
afterwards Reception

Tuesday, October 17

09:30 - 10:30 Ben Davison: Kac-Moody Lie algebras as BPS Lie algebras
10:30 - 11:00 Group photo and coffee break
11:00 - 12:00 Penghui Li: Analytic atlas for stack of semistable bundles on elliptic curve and elliptic character sheaves
12:00 - 13:20 Lunch break
13:20 - 14:20 Spela Spenko
14:20 - 15:20 Michael McBreen: Quantum Connections vs Quantizations for Symplectic Dual Spaces
15:30 - 17:30 Hirzebruch Birthday Colloquium (at MPIM)
17:50 - 18:50 Michel Van den Bergh (at HIM!)
19:00 - Conference dinner

Abstracts

Gwyn Bellamy: Graded algebras admitting a triangular decomposition

The goal of this talk is to describe the representation theory of finite dimensional graded algebras A admitting a triangular decomposition (in much the same flavour as the enveloping algebra of a semi-simple Lie algebra admits a triangular decomposition). The examples to keep in mind are restricted rational Cherednik algebras, restricted enveloping algebras and hyperalgebras. We exploit the fact that the category of graded modules for such an algebra is a highest weight category. This allows us to prove two key results. First that the degree zero part A0 of the algebra is cellular, and secondly a canonical subquotient of our highest weight category provides a highest weight cover of A0-mod. This is based on joint work with U. Thiel.

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Ben Davison: Kac-Moody Lie algebras as BPS Lie algebras

In this talk I will explain how a surprising perverse filtration on the Kontsevich-Soibelman cohomological Hall algebra for a quiver with potential (Q,W) allows us to define the BPS Lie algebra for (Q,W), categorifying refined BPS invariants.  This Lie algebra comes with a cohomological grading, and for a given quiver Q, there is a quiver with potential for which the degree zero part of the BPS Lie algebra is the positive half of the Kac Moody Lie algebra for the real subquiver of Q.

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Tina Kanstrup: Braid group actions, matrix factorizations and link invariants

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.

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Allen Knutson: Schubert calculus via degenerate R-matrices

In 1999, based on a rule for Schubert calculus on 1-step flag manifolds (i.e. Grassmannians), I conjectured a rule for d-step flag manifolds. This was wrong already for d=3 (but conjecturally corrected by Buch), but proven for d=2 in 2014 by Buch-Kresch-Purbhoo-Tamvakis. I'll explain how to derive these puzzle rules from the R-matrices of certain minuscule representations of A2, D4, E6, improving on old proofs while giving rules for several problems that had not yet even had conjectures (K-theory for 2-step and 3-step, equivariant K-theory for 2-step). This paper arXiv:1706.10019 is joint with Paul Zinn-Justin.

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Martina Lanini: Sheaves on the alcoves and modular representations

I'll report on a joint project with Peter Fiebig. The project is meant to give a new perspective on the problem of calculating irreducible characters of reductive algebraic groups in positive characteristics. Given a finite root system R and a field k we introduce an exact category C of sheaves on the partially ordered set of alcoves associated with R, and we show that the indecomposable projective objects in C encode the aforementioned characters.

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Penghui Li: Analytic atlas for stack of semistable bundles on elliptic curve and elliptic character sheaves

Let G be a simply connected algebraic group, we define an analytic atlas of the moduli stack of semistable G-bundles on elliptic curves. Each chart in the atlas is isomorphic to an open substacks of the adjoint quotient of (explicitly described) reductive subgroups of G. This atlas preserves sheaves with nilpotent singular support under pullback, and we present the category of elliptic character sheaves as a limit (over the diagram of faces of the affine alcove of G) of the categories of character sheaves on the reductive subgroups defined above. We also discuss the relation with Betti Langlands program initiated by Ben-Zvi and Nadler (in genus 1). This is a joint work with D.Nadler.

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Michael McBreen: Quantum Connections vs Quantizations for Symplectic Dual Spaces

I will describe joint work with Joel Kamnitzer and Nick Proudfoot relating the quantum connection of a symplectic resolution to the quantization of the symplectic dual resolution.

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Yoshiki Oshima: The orbit method and characters of representations for real reductive groups

The orbit method was introduced by Kirillov and relates coadjoint orbits and unitary representations of Lie groups.  For nilpotent groups it establishes an exact correspondence between the unitary representations and the coadjoint orbits.  Although such correspondence is not so perfect for reductive groups, it still provides a general principle for the study of unitary representations.  In this talk we study distribution characters of singular unitary representations of real reductive groups from the viewpoint of orbit philosophy. This talk is based on a joint work with Benjamin Harris.

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Olivier Schiffmann: Counting cuspidals for quivers and curves

We consider the problem of determining the minimal number of generators in each degree for the (entire) Hall algebras of curves and quivers. This is strongly related to Kac polynomials, cohomological Hall algebras and, in the case of quivers, conjecturally related to the Lie algebras introduced by Maulik and Okounkov by means of stable enveloppes in Nakajima quiver varieties.

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Peng Shan: On the cohomology of Calogero-Moser spaces

I will report a recent joint work with Cédric Bonnafé in which we compute equivariant cohomology of smooth Calogero-Moser spaces and some related symplectic resolutions.

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Wolfgang Soergel: Koszul duality for real groups

I will try to explain how advances in the construction of motivic sheaves allow to give a much more explicit form to my old conjectures on Langlands philosophy and Koszul duality, which were among others motivated by a remark of Victor Ginzburg. This is in parts joint work in progress with Matthias Wendt and Rahbar Virk and in part the outcome of discussions with Joseph Bernstein.

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Yaping Yang: A geometric construction of affine quantum groups

We will talk about a construction of affine quantum groups using cohomological Hall algebras using a generalized cohomology theory. In my talk, I will focus on two examples:
 1. We use the Morava K-theory to construct a family of new quantum groups parametrized by a prime number and a positive integer. Those quantum groups are expected to be related to Lusztig’s 2015 reformulation of his conjecture from 1979 on character formulas for algebraic groups over a field of positive characteristic.
2. We use the equivariant elliptic cohomology to establish a sheafified elliptic quantum group for any symmetric Kac-Moody Lie algebra. The rational sections give the algebra of elliptic R-matrix. I will also explain the relation between the sheafified elliptic quantum group and a global loop Grassmannian over an elliptic curve.
This talk is based on my joint work with Gufang Zhao.

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