Camp Style Seminar 2015:Repsentation theory and Related Topics

How to go: We will pick up invited speakers who are staying at the Comfort Hotel Chubu International Airport at the hotel in the morning of February 16.
If you are coming on your own there will be a free bus from West Exist of Toyohashi station to Irako View Hotel at 12:40 on Monday, February 16. It will take about 2 hours from Toyohashi station to the hotel. You can also take a public bus from # 1 bust stop of East Exist of Toyohashi station to Irako Misaki (the final bus stop), see the table table for weekdays (only in Japansese). It will cost ¥1,490. They will pick you up if you call the hotel from the final bus stop, Irako Misaki (from 8:00 to 18:30 only).
For returen there will be a free bus from the hotel to Toyohashi station at 14:00 on Friday, February 20.

	15:00-15:30	15:30--16:30	16:45--17:45	18:00--19:00	19:00-19:30
Mon, Feb. 16	registration	Bellamy	Yanagida	McGerty	Free discussion

	9:00--10:00	10:15--10:45	11:00--12:00	12:00-13:00	13:00-14:30	14:30--15:00	15:15--16:15	16:30--17:00	17:00-19:00
Tue, Feb. 17	Baumann	Tsuchioka	Riche	Lunch	Free discussion	Lanini	Hikita	Oshima	Free discussion
Wed, Feb. 18	Sevastyanov	Naoi	Moreau	Lunch	Free discussion	Free discussion	Free discussion	Free discussion	Free discussion
Thu, Feb. 19	Oblomkov	Kimura	Arinkin	Lunch	Free discussion	Kodera	Guay	Wada	Free discussion

	9:00--10:00	10:00--10:30	10:30--11:30	11:30--12:30	12:30--14:00
Fri, Feb. 20	Kanstrup	break	Juteau	Lunch	Free discussion

Let G be a complex reductive group. Consider categories equipped with smooth (sometimes called strong) action of G. Natural and important examples of such categories arise from geometry: if X is a variety equipped with an action of G (for instance, X=G itself, or X=G/B is the flag space of G), then the category of D-modules on X carries a smooth action of G. We view such categories as (smooth) categorical representations of G.
The theory of smooth categorical representations of G is similar to the representation theory of a reductive group over a finite field, I will discuss this similarity in my talk. The surprising twist (and the main result of this talk) is that the theory of smooth categorical representations is simpler than its classical counterpart: there are no cuspidal representations!

(This is work in progress joint with Gaussent and Littelmann.)
Let G be a connected reductive algebraic group over the field of complex numbers. The geometric Satake correspondence (due to Lusztig, Beilinson-Drinfeld, Ginzburg, and Mirković-Vilonen) provides a tensor equivalence between a category of perverse sheaves on the affine Grassmannian $Gr$ of the Langlands dual of $G$, endowed with a convolution product, and the category of finite-dimensional representations of $G$. Specifically, given a dominant weight $\lambda$ for $G$, the intersection homology of the closure of the Schubert cell $Gr^\lambda\subset Gr$ can be endowed with the structure of a $G$-module of highest weight $\lambda$. Further, one can define cycles in $\overline{Gr^\lambda}$, the so-called Mirković-Vilonen cycles, whose fundamental classes form a basis in the intersection homology. A similar construction allows to construct geometrically the tensor products of simple $G$-modules: the Schubert varieties $\overline{Gr^\lambda}$ must be replaced by convolution varieties and the Mirković-Vilonen cycles must be replaced by cycles defined by Goncharov and Shen. We will study the relationship between the basis formed by Goncharov and Shen's cycles and the basis given by tensor products of Mirković-Vilonen cycles.

If V is a symplectic vector space and G a finite subgroup of Sp(V), then the quotient singularity V/G is a very interesting object to study, both from the geometric and representation-theoretic point of view. One of the motivational problems in trying to understand the singularities of V/G is that of deciding whether V/G admits a symplectic resolution or not. More generally, one can ask how many symplectic resolutions it admits. The goal of this talk is to explain how one can count the number of symplectic resolutions of V/G. I'll present an explicit formula for this number in terms of the dimension of a certain Orlik-Solomon algebra. The key to deriving this formula is to relate the resolutions of V/G to the Calogero-Moser deformations, where one can use the representation theory of symplectic reflection algebras.

Nicolas Guay, Deformed double current algebras associated to complex semisimple Lie algebras.

Almost ten years ago, deformed double current algebras for $\mathfrak{sl}_n$ appeared in the study of Schur-Weyl duality for rational Cherednik algebras of the symmetric group. For an arbitrary complex semisimple Lie algebra, they were defined a couple of years later as degenerate forms of affine Yangians. They have appeared recently in on-going work of V. Toledano-Laredo and Y. Yang on the elliptic Casimir connection. I will give an overview of a few results known about deformed double current algebras and I will discuss some possible directions for studying their representations, based in part on known results about quantum toroidal algebras and Yangians.

Tatsuyuki Hikita, An algebro-geometric realization of the cohomology ring of Hilbert scheme of points in the affine plane.

We show that the cohomology ring of Hilbert scheme of $n$-points in the affine plane is isomorphic to the coordinate ring of $\mathbb{G}_{m}$-fixed point scheme of the $n$-th symmetric product of $\mathbb{C}^{2}$ for a natural $\mathbb{G}_{m}$-action on it. This result can be seen as an analogue of a theorem of DeConcini, Procesi and Tanisaki on a description of the cohomology ring of Springer fiber of type A.

This is a report on joint work with Armin Gusenbauer, Stephen Griffeth, and Martina Lanini. We give necessary conditions for finite dimensionality of a simple lowest weight module for the rational Cherednik algebra of a complex reflection group, and for the existence of a non-zero map between standard modules. The latter criterion involves a generalization of the combinatorics of standard Young tableaux to arbitrary complex reflections groups. This uses a parabolic degeneration procedure for Cherednik algebra modules, alternative to Bezrukavnikov-Etingof parabolic restriction.

Tina Kanstrup, Quasi-coherent Hecke categories and affine braid group actions.

We propose a geometric setting leading to categorical braid group actions.
First we consider the quasi-coherent Hecke category QCHecke(G,B) for a reductive group G with a Borel subgroup B. We show that a monoidal action of QCHecke(G,B) on a triangulated category gives rise to a categorification of degenerate Hecke algebra representation known as Demazure Descent Data.
Next we replace the group G by the derived group scheme LG of topological loops with values in G and consider QCHecke(LG,LB). A monoidal action of the category QCHecke(LG,LB) gives rise to a categorical action of the affine Braid group.
Trying to avoid heavy derived algebraic geometry methods, we present an example of the construction above in classical algebro-geometric terms. For a G-variety $X$, we construct a Braid group action on a category of equivariant matrix factorizations on the product of $T^*X$ and the Grothendieck variety for the Lie algebra of G. The potential for matrix factorizations is provided by the moment map.

Yoshiyuki Kimura, Remarks on quantum unipotent subgroups and the dual canonical basis.

Quantum unipotent subgroup is the quantum coordinate ring of the unipotent subgroup associated with a Weyl group element of a symmetrizable Kac-Moody group. By the works of Geiss-Leclerc-Schroer and Goodearl-Yakimov, it is known that it has a quantum cluster algebra structure. In this talk, I explain the opposite pro-unipotent subgroup associated with a given Weyl group element and its compatibility with the dual canonical basis and the multiplicative property between the dual canonical basis element in the quantum unipotent subgroups and the one in the opposite.

We compare actions of Yangians of finite and affine type A on the Fock space. The former is constructed by Uglov using Drinfeld correspondence, while the latter is by Varagnolo via quiver variety.

Introduced in 2010 by E. Feigin, degenerate flag varieties are degenerations of flag manifolds. It has been proven that, in type A and C, they share many properties with Schubert varieties. In this talk I will discuss joint work with Cerulli Irelli, where we prove a surprising fact about degenerate flags.

Kevin McGerty, KN stratifications, D-modules and a modified hyper-Kahler Kirwan map.

We will discuss a recollement for D-modules on stacks equipped with a certain kind of stratification which arises naturally in GIT. Applications to the cohomology of hyper-Kahler quotients such as quiver varieties will be given. This is joint work with Tom Nevins.

Anne Moreau, Motivic stringy invariants for spherical varieties and applications.

In this talk, we will be interested in the motivic integral over the arc space of a complex Q-Gorensein spherical G-variety X where G is a reductive connected group. We give a formula for the stringy E-function of X in term of its colored fan. As applications, we obtain a smoothness criterion for locally factorial horospherical varieties and we conjecture that this criterion still holds for any locally factorial spherical variety. We also compute the Poincaré series of the weighted graded Stanley-Reisner ring for a Q-Gorenstein horospherical variety $X$, which allows to describe the cohomology ring $H^*(X,C)$ is some particular cases. All this is based on joint (past and current) works with Victor Batyrev.

Minimal affinizations are a class of finite-dimensional simple modules over a quantum loop algebra. In this talk, I will introduce a Jacobi-Trudi type formula for the characters of minimal affinizations in classical types. This formula is proved by studying the graded limits (variants of classical limits) of minimal affinizations. If time permits, I would also like to give a sketch of the proof.

Alexei Oblomkov, Cohomology of the homogeneous Hitchin fibers and representation theory of Cherednik algebras.

The talk is based on the joint work with Z. Yun.
I will explain a geometric construction of the action of the rational Cherednik algebra on the cohomology of the homogeneous Hitchin fiber for the Hitchin system on the weighted projective line. The perverse and cohomological filtrations on the cohomology have a natural representation theoretic interpretation which will be explained. In type $A$ case, the ring structure of the cohomology will be presented.

The restriction of the oscillator representation of the metaplectic group to a reductive dual pair gives a correspondence between representations of different reductive groups, which is called the Howe duality. We study the representations of real reductive groups which correspond to holomorphic representations by the Howe duality using D-module realization.

Simon Riche, Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirkovic-Vilonen conjecture.

In this talk I'll explain the construction (obtained in a joint work with Carl Mautner) of an equivalence of categories relating tilting objects in the heart of Bezrukavnikov's "exotic t-structure" on the derived category of equivariant coherent sheaves on the Springer resolution of a connected reductive group over a field k of good characteristic, and Iwahori-constructible parity sheaves on the affine Grassmannian Gr of the Langlands dual group, with coefficients in k. This construction allows in particular to obtain a modular version of an equivalence due to Arkhipov-Bezrukavnikov-Ginzburg in characteristic 0, and to prove the last missing piece for the proof of the Mirkovic-Vilonen conjecture on stalks of standard spherical perverse sheaves on Gr in the expected generality.

Alexey Sevastyanov, A proof of De Concini-Kac-Procesi conjecture and Lusztig's partition.

In 1992 De Concini, Kac and Procesi observed that isomorphism classes of irreducible representations of a quantum group at odd primitive root of unity m are parameterized by conjugacy classes in the corresponding algebraic group G. They also conjectured that the dimensions of irreducible representations corresponding to a given conjugacy class $O$ are divisible by $m^{1/2 dim O}$. In this talk I shall outline a proof of an improved version of this conjecture and derive some important consequences of it related to q-W algebras.

A key ingredient of the proof are transversal slices S to the set of conjugacy classes in G. Namely, for every conjugacy class O in G one can find a special transversal slice S such that O intersects S and dim O=codim S. The construction of the slice utilizes some new combinatorics related to invariant planes for the action of Weyl group elements in the real reflection representation. The condition dim O=codim S is checked using some new mysterious results by Lusztig on intersection of conjugacy classes in algebraic groups with Bruhat cells.

Shunsuke Tsuchioka, On graded Cartan invariants of symmetric groups and Hecke algebras.

We consider graded Cartan matrices of the symmetric groups and the Iwahori-Hecke algebras. These matrices may also be interpreted as Gram matrices of the Shapovalov form on sums of weight spaces of a basic representation of an affine quantum group. We present a conjecture predicting the invariant factors of these matrices and proving that it gives the correct invariants when one specializes or localizes the ring $\mathbb{Z}[v,v^{-1}]$ in certain ways. This generalizes Evseev's theorem which settled affirmatively the K\"ulshammer-Olsson-Robinson conjecture that predicts the Cartan invariants of the Iwahori-Hecke algebras. This is a joint work with Anton Evseev.

Shintaro Yanagida, Quantum toroidal algebras and motivic Hall algebras of elliptic singular fibers.

We consider the Ringel-Hall algebra of coherent sheaves over the trees or cycles of projective lines. The Drinfeld double of the composition subalgebra of this Hall algebra inherits automorphisms induced from equivalences of the associated derived category. We construct non-trivial automoprhisms of the Drinfeld double in the case of Kodaira singular fibers of elliptic surfaces using the relative Fourier-Mukai transforms and the motivic version of the Hall algebra. These non-trivial automorphisms coincide with the one constructed by Miki for the quantum toroidal algebras in the case of type A.

A cyclotomic $q$-Schur algebra $\mathscr{S}_{n,r}$, which has parameters $q, Q_0,Q_1,\dots,Q_{r-1}$, is a quasi-hereditary cover of the cyclotomic Hecke algebra of type $G(r,1,n)$ (i.e. Ariki-Koike algebra) in the sense of Rouquier. In the case where $r=1$, $\mathscr{S}_{n,1}$ is a classical $q$-Schur algebra, and it is known that $\mathscr{S}_{n,1}$ is a quotient of the quantum group $U_q (\mathfrak{gl}_m)$. However, in the case where $r>1$, $\mathscr{S}_{n,r}$ is not a quotient of a quantum group.

In this talk, we introduce a Lie algebra $\mathfrak{g}$ with parameters $Q_1,\dots, Q_{r-1}$ and an associative algebra $\mathcal{U}$ with parameters $q, Q_1,\dots,Q_{r-1}$. Then, $\mathscr{S}_{n,r}$, in the case where $q$ is generic (resp. $q=1$), is realized as a quotient of $\mathcal{U}$(resp. $U(\mathfrak{g})$). We see that $\mathfrak{g}$ is a deformation of the current Lie algebra of $\mathfrak{gl}_m$, and we see that $\mathcal{U}$ is "a $q$-analogue" of $U(\mathfrak{g})$. We also discuss some properties of them.

Organizing committee: Hiraku Nakajima, Syu Kato, Tomoyuki Arakawa

Contact: arakawa at kurims.kyoto-u.ac.jp

Repsentation theory and Related Topics

(Camp Style Seminar)

Feb. 16, 2015--Feb. 20, 2015