## Workshop Geometry and representation theory at the interface of Lie algebras and quivers 10 - 14 September 2018 Ruhr-Universitaet Bochum Organizers: Evgeny Feigin (Moscow), Markus Reineke (Bochum)

The workshop intends to bring together experts in geometry and representation theory working at the interface of Lie algebras and quivers.

Topics to be discussed include: representations of current and affine algebras, geometry of affine and semi-infinite flag varieties, degenerate flag varieties and quiver Grassmannians, Hall algebras, spherical varieties, toric degenerations of flag varieties, Newton-Okounkov polytopes.

## Invited speakers

Ivan Arzhantsev (Moscow)
Alexander Braverman (Toronto)
Giovanni Cerulli Irelli (Rome)
Stephanie Cupit-Foutou (Bochum)
Ghislain Fourier (Hannover)
Victor Ginzburg (Chicago)
David Hernandez (Paris)
Syu Kato (Kyoto)
Valentina Kiritchenko (Moscow)
Martina Lanini (Rome)
Peter Littelmann (Cologne)
Satoshi Naito (Tokyo)
Daniel Orr (Virginia Tech)
Konstanze Rietsch (London)
Mark Shimozono (Virginia Tech)
Yuri Tschinkel (New York)
Lauren Williams (Berkeley)
Oksana Yakimova (Cologne)

## Schedule

Monday 10 September:

10:00 - 11:00 : Peter Littelmann: Semitoric degenerations via Newton-Okounkov bodies and Standard Monomial Theory
11:30 - 12:30 : Yuri Tschinkel: Height zeta functions

14:00 - 15:00 : Ivan Arzhantsev: Infinite transitivity, finite generation and Demazure roots
15:30 - 16:30 : Oksana Yakimova: Quantisation and nilpotent limits of Mishchenko--Fomenko subalgebras
17:00 - 18:00 : Martina Lanini: Cohomology of the flag variety under PBW degenerations

Tuesday 11 September:

10:00 - 11:00 : David Hernandez: Grothendieck ring isomorphims, mutated quivers and Kazhdan-Lusztig polynomials
11:30 - 12:30 : Giovanni Cerulli Irelli: Specialization of Homology for quiver Grassmannians

15:00 - 16:00 : Alexander Braverman: tba
16:30 - 17:30 : Syu Kato: Loop structure on equivariant $K$-theory of semi-infinite flag manifolds

Wednesday 12 September:

10:00 - 11:00 : Mark Shimozono: Quiver Kostka-Shoji polynomials
11:30 - 12:30 : Daniel Orr: Quiver Hall-Littlewood functions via vertex operators

Afternoon: Excursion to German Mining Museum

Thursday 13 September:

10:00 - 11:00 : Konstanze Rietsch: Newton-Okounkov bodies for Grassmannians I
11:30 - 12:30 : Lauren Williams: Newton-Okounkov bodies for Grassmannians II

15:00 - 16:00 : Ghislain Fourier: Degenerate Schubert varieties in type A
16:30 - 17:30 : Valentina Kiritchenko: Newton-Okounkov polytopes of flag and Bott-Samelson varieties

Friday 14 September:

09:00 - 10:00 : Stephanie Cupit-Foutou: Momentum polytopes of projective complex spherical varieties
10:30 - 11:30 : Satoshi Naito: Pieri-Chevalley formula in the equivariant K-theory of semi-infinite flag manifolds
12:00 - 13:00 : Victor Ginzburg: Differential operators on G/U and the Gelfand-Graev action

## Titles and abstracts

Ivan Arzhantsev: Infinite transitivity, finite generation and Demazure roots.
Abstract: A complex affine algebraic variety X of dimension at least two is called flexible if the subgroup SAut(X) of the automorphism group Aut(X) generated by all one-parameter subgroups acts m-transitively on the smooth locus of X for all positive integer m. It is known that any nondegenerate affine toric variety is flexible.
We prove that if such a toric variety X is smooth in codimension two  then one can find a subgroup of SAut(X) generated by a finite number of one-parameter unipotent subgroups which has the same transitivity property. In fact, three such subgroups are sufficient when X is an affine space. Our proofs are based on the study of closures of some infinite dimensional groups. This approach leads to natural constructions and questions on Lie algebras of such groups.
This is a joint work with Karine Kuyumzhiyan and Mikhail Zaidenberg.

Giovanni Cerulli Irelli: Specialization of Homology for quiver Grassmannians
Abstract: Lanini and Strickland showed that there is an injective ring homomorphism from the Borel-Moore homology of a PBW degeneration of the complete ag variety to the Borel-Moore homology of the complete ag variety itself. Since those varieties are particular linear degenerations of ag varieties, it is hence natural to ask if the same statement remains true for two representations M and M’ such that $M\leq_{\deg} M’$. We give a positive answer to this question for quivers of type A. Our approach uses the fact that minimal degenerations are given by generating short exact sequences. This is a joint work in progress with F. Esposito, X. Fang and G. Fourier.

Stephanie Cupit-Foutou: Momentum polytopes of projective complex spherical varieties.
Abstract: Momentum polytopes of polarized complex algebraic varieties (X, L) acted on by a reductive algebraic group G are rational convex polytopes; they encode the asymptotic behavior of the irreducible G-modules occurring in the total section ring of (X, L). This talk deals with the case of spherical varieties X, that are varieties whose all section rings are multiplicity-free G-modules. Our mail goal will be to give a characterization of their momentum polytopes and sketch some applications of this characterization -- some of these results have been obtained jointly with G. Pezzini and B. Van Steirteghem.

Ghislain Fourier: Degenerate Schubert varieties in type A
Abstract: I’ll briefly recall PBW degenerations of flag varieties and the various points of view on them, including the identification with Schubert varieties in double rank flag varieties due to Cerulli-Irelli, Lanini and Littelmann. We consider PBW degenerate Schubert varieties of type A and try to obtain similar results here. For this, I‘ll introduce rectangular elements in the symmetric group and show that the degenerate Schubert variety associated to a rectangular element is indeed a Schubert variety in a partial flag variety of the same type with larger rank. Moreover, the degenerate Demazure module associated to a rectangular element is isomorphic to the Demazure module for this particular Schubert variety of larger rank. This is joint work with Xin Fang and Rocco Chirivi.

Victor Ginzburg: Differential operators on G/U and the Gelfand-Graev action.
Abstract: Let G be  a complex semisimple group and U its maximal unipotent subgroup. We study the algebra D(G/U) of algebraic differential operators on G/U and also its quasi-classical counterpart: the algebra of regular functions on the cotangent bundle. A long time ago, Gelfand and Graev have constructed an action of the Weyl group on D(G/U) by algebra automorphisms. The Gelfand-Graev construction was not algebraic, it involved the Hilbert space L^2(G/U) in an essential way. We give  a new algebraic construction of the  Gelfand-Graev action, as well as its quasi-classical counterpart. Our approach is based on Hamiltonian reduction and involves the ring of Whittaker differential operators on G/U, a twisted analogue of  D(G/U). Our main theorem  has an interpretation, via Geometric Satake,  in terms of spherical perverse sheaves on the affine Grassmanian for the Langlands dual group.

David Hernandez: Grothendieck ring isomorphims, mutated quivers and Kazhdan-Lusztig polynomials
Abstract:Quantum Grothendieck rings are natural t-deformations of representations rings of quantum affine algebras. They are known to have a structure of a quantum cluster algebra. Using distinguished equivalences of corresponding quivers, we establish ring isomorphism between quantum Grothendieck rings in types A and B. Combining we recent results of Kashiwara-Kim-Oh, we prove for the corresponding categories in type B a conjecture formulated by the speaker in 2002 : the multiplicities of simple modules in standard modules are given by the evaluation of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive (joint work with Hironori Oya; supported by the ERC Grant Agreement number 647353 Qaffine).

Syu Kato: Loop structure on equivariant $K$-theory of semi-infinite flag manifolds
Abstract: In his 1994 ICM address, Givental posed a program to equip the homology group of a loop space with an operation that gives rise to the multiplication of the quantum cohomology.
In our previous paper with Naito and Sagaki, we proposed a definition of the equivariant $K$-group of semi-infinite flag manifold, that is one of the algebro-geometric models of the loop space of the flag manifold.
In this talk, we explain that our group should be in some sense the correct one by exhibiting explicit isomorphisms between our $K$-group and the equivariant $K$-group of affine Grassmanians and the equivariant quantum $K$-group of the flag manifold. In particular, we have a posteriori definition of the product structure of the equivariant $K$-group of flag manifolds that realizes Givental’s program in our setting, that is quite simple.

Valentina Kiritchenko: Newton-Okounkov polytopes of flag and Bott-Samelson varieties
Abstract: Polytopes from representation theory such as string and FFLV polytopes can be realized as Newton-Okounkov convex bodies of flag varieties for various geometric valuations. More generally, the same valuations can be used to construct Newton-Okounkov convex bodies of Bott-Samelson resolutions. I will discuss relationship between the Newton-Okounkov polytopes of flag and Bott-Sameslon varieties. In the case of flag varieties for classical groups, I will define a valuation that seems to be especially suitable for explicit computation of Newton-Okounkov polytopes. In type A, Newton-Okounkov polytopes of Bott-Samelson varieties for this valuation are Minkowski sums of FFLV polytopes.

Martina Lanini: Cohomology of the flag variety under PBW degenerations
Abstract: Given a complex algebraic variety and a proper flat degeneration of it (over A^1), there is an induced homomorphism from the cohomology of the latter to the cohomology of the original variety. Such a homomorphism can in general fail to be surjective or injective. In my talk, I will discuss joint work with Elisabetta Strickland, in which we show that in the case of PBW degenerations of a (type A) flag variety, the induced homomorphism between cohomologies is surjective. PBW degenerations have been recently introduced by Cerulli Irelli, Fang, Feigin, Fourier and Reineke, and proven to have the nice property to be isomorphic to Schubert varieties. Our result shows once more that these degenerations are extremely well behaved.

Peter Littelmann: Semitoric degenerations via Newton-Okounkov bodies and Standard Monomial Theory
Abstract: Sequences of Schubert varieties, contained in each other and successively of codimension one, naturally lead to valuations on the field of rational functions of the flag variety. By taking the minimum over all these valuations, one gets a quasi valuation which leads to a semitoric degeneration of the flag variety. We show that this semitoric  degeneration is strongly related to the Standard Monomial Theory on flag varieties as originally initiated by Seshadri, Lakshmibai and Musili. This is work in progress jointly with Rocco Chirivi and Xin Fang.

Satoshi Naito: Pieri-Chevalley formula in the equivariant K-theory of semi-infinite flag manifolds
Abstract: We give a Pieri-Chevalley formula for anti-dominant weights in the torus-equivariant K-group of a semi-infinite flag manifold, which describes the (tensor) product of the class of a line bundle with the class of the structure sheaf of a semi-infinite Schubert variety, in terms of the semi-infinite analog of Lakshmibai-Seshadri paths. As an application, we obtain a Monk formula in the K-group above, which describes the multiplication by the class of the structure sheaf of a semi-infinite Schubert variety of codimension one. On the basis of the isomorphism between the K-group above and the torus-equivariant (small) quantum K-group of an ordinary flag manifold, which has recently been established by Syu Kato, our results yield an explicit description of the quantum product by a line bundle associated to an anti-dominant fundamental weight; in particular, in type A, we can verify a conjectural Monk formula (presented by Lenart and Postnikov) in the quantum K-theory of a flag manifold. This talk is based on a joint work with D. Sagaki and D. Orr.

Daniel Orr: Quiver Hall-Littlewood functions via vertex operators
Abstract: In joint work with Mark Shimozono, we introduce quiver analogues of Hall-Littlewood symmetric functions. For a given quiver with r vertices, these belong to the r-fold tensor power of the algebra of symmetric functions, with coefficients in a field containing one parameter for each arrow in the quiver. Our functions include the ordinary Hall-Littlewood functions (Jordan quiver) and Shoji functions (cyclic quiver) as special cases, as well as their parabolic versions.
The quiver Hall-Littlewood functions can be regarded as truncations'' of equivariant Euler characteristics of vector bundles on Lusztig's convolution diagrams; this generalizes the interpretation of Kostka polynomials, due to R. Brylinski, via Euler characteristics of line bundles on the cotangent bundle of the flag variety. Of particular interest are the transition coefficients, which we call quiver Kostka-Shoji polynomials, between the quiver Hall-Littlewood functions and the tensor Schur functions.
In this talk, I will explain the construction of quiver Hall-Littlewood functions based on vertex operators, generalizing those of Garsia and Jing from the ordinary Hall-Littlewood case. In the special cases mentioned above, these vertex operators arise naturally in representations of quantum toroidal algebras of type GL. In general they give rise to a representation of the K-theoretic Hall algebra of the quiver on the tensor power of symmetric functions.

Konstanze Rietsch, Lauren Williams: Newton-Okounkov bodies for Grassmannians (I and II)
Abstract: In recent work (arXiv:1712.00447) we use the X-cluster structure on the Grassmannian and the combinatorics of planar bicolored graphs to associate a Newton-Okounkov body to each X-cluster. This gives, for each X-cluster, a toric degeneration of the Grassmannian. We also describe the Newton-Okounkov bodies explicitly using mirror symmetry: we show that they are polytopes whose facets can be read off from A-cluster expansions of the superpotential of Marsh-R. And we give a combinatorial formula for the lattice points of the Newton-Okounkov bodies, which has a surprising interpretation in terms of quantum Schubert calculus.

Mark Shimozono: Quiver Kostka-Shoji polynomials
Abstract: In joint work with Dan Orr, we study the Euler characteristics of global sections applied to the twist by certain vector bundles, of Lusztig's convolution diagram, which is itself a vector bundle on a product of partial flag varieties, one for each quiver node. We give a conjecture for higher vanishing. The case of the cyclic quiver was studied by Finkelberg and Ionov and connects with earlier work of Shoji on Green's functions for complex reflection groups. For a single loop quiver the higher vanishing conjecture goes back to Broer in the 1990s. We give conjectures for positive formulas for these polynomials in the case of the cyclic and single path equioriented quivers.

Yuri Tschinkel: Height zeta functions
Abstract: I will discuss geometric and arithmetic constructions arising in the study of the distribution of rational points on orbits of linear algebraic groups.

Oksana Yakimova: Quantisation and nilpotent limits of Mishchenko--Fomenko subalgebras
Abstract: Based on a joint project with A. Molev.
Let g be a reductive Lie algebra, \mu be a linear function on g, and A_\mu \subset S(g) be the shift of argument, also known as Mishchenko--Fomenko, subalgebra associated with \mu. Then  A_\mu is Poisson-commutative. The task of lifting A_\mu  to U(q) constitutes Vinberg's quantisation problem". Various methods for solving it have been found. We will discuss difficulties of the non-regular case, where the coadjoint orbit through \mu is not of the maximal dimension. Also the role of the symmetrisation map in the quantisation of MF-subalgebras will be explained. The symmetrisation map commutes with taking limits thus allowing one to quantise various limits of MF-subalgebras.

## Participants

Susumu Ariki (Osaka)
Ivan Arzhantsev (Moscow)
Roman Avdeev (Moscow)
Ravinder Bhimarti (Chennai)
Leon Barth (Bochum)
Maria Bertozzi (Bochum)
Magdalena Boos (Bochum)
Alexander Braverman (Toronto)
Giovanni Cerulli Irelli (Rome)
Narasimha Chary (Bonn)
Yunhyung Cho (Seoul)
Stephanie Cupit-Foutou (Bochum)
Xin Fang (Cologne)
Evgeny Feigin (Moscow)
Ghislain Fourier (Aachen)
Hans Franzen (Bonn)
Victor Ginzburg (Chicago)
Mikhail Gorsky (Bielefeld)
David Hernandez (Paris)
Syu Kato (Kyoto)
Valentina Kiritchenko (Moscow)
Fabian Korthauer (Bochum)
Deniz Kus (Bochum)
Martina Lanini (Rome)
Peter Littelmann (Cologne)
Ievgen Makedonsky (Moscow)
Igor Makhlin (Moscow)
Joanna Meinel (Stuttgart)
Lang Mou (UC Davis)
Satoshi Naito (Tokyo)
Daniel Orr (Vtech)
Alexander Panov (Samara)
Li Penghui (IST Austria)
Alexander Pütz (Bochum)
Andriy Regeta (Cologne)
Markus Reineke (Bochum)
Konstanze Rietsch (London)
Shiquan Ruan (Bielefeld)
David Sabonis (Munich/Copenhagen)
Mark Shimozono (Vtech)
Christian Steinert (Cologne)
Yuri Tschinkel (New York)
Thorsten Weist (Wuppertal)
Lauren Williams (Berkeley)
Oksana Yakimova (Cologne)

## Practical information

All talks of the workshop will take place in building SSC on the campus close to the entrance of the campus.
Arriving at the subway station, cross the bridge to the campus, then building SSC is located on the left.

From the city centre / Bochum main station, the campus can be conveniently reached by subway line U35 "CampusLinie", direction Bochum Hustadt, which departs every five minutes and brings you to the campus in less than 10 minutes.

Here is a list of hotels, of various price classes, all located in the city centre:

Hotel Claudius
Ibis City
Ibis Zentrum
Landesspracheninstitut Bochum
Mercure Hotel Bochum
Art Hotel Tucholsky
Youth hostel Bochum

The workshop is funded by the DFG/RSF project "Geometry and representation theory at the interface of Lie algebras and quivers" and the DFG project CRC/TRR 191 "Symplectic Structures in Geometry, Algebra and Dynamics"