Representation Theory of Algebraic Groups and Quantum Groups 06

Representation Theory
of Algebraic Groups and Quantum Groups '10

August 2 - 6, 2010

Room 509, Graduate School of Mathematics, Nagoya Univesity

to Menu


Tomoyuki Arakawa (Nara Women's University)
Rationality and smoothness of W-algebras

In my talk I am going to discuss the rationality and the C_2- cofinitness of W-algebras and their relations with representations of Kac-Moody algebras.

Roman Bezrukavnikov (MIT)
Canonical bases and modular representations

I will briefly review the main ingredients of the proof of Lusztig's conjectures which generalize Kazhdan-Lusztig conjectures to representations of Lie algebras in positive characteristic (joint with Mirkovic and partly with Rumynin). I will then describe a joint project with Okounkov which generalizes the picture to representations of rational Cherednik algebras of type A.

Jonathan Brundan (University of Oregon)
Rational representations of GL(m|n) via Schur-Weyl duality

I will explain an approach to the rational (not just polynomial) representation theory of the complex supergroup GL(m|n) via a generalized Schur-Weyl duality. This allows the endomorphism algebra of a minimal projective generator to be computed explicitly: it turns out to be a limiting version of Khovanov's diagram algebra which is Koszul. This is joint work with Catharina Stroppel.

Boris Feigin (Independent University of Moscow)
Representation theory of some version of quantum gl infinity

In my talk I will explain how the representations of elliptic W-algebras naturally appear when you study the moduli spaces of bundles on projective plane. Such connection with algebraic geometry gives us some new and interesting things from the representation theory of W-algebras.

Peter Fiebig (Erlangen University)
On the critical level representations of affine Kac-Moody algebras

We review recent joint work with Tomoyuki Arakawa on the category O of an affine Kac-Moody algebra at the critical level. We are especially interested in the subcategory of restricted representations. Lusztig anticipated a connection to the representation theory of a small quantum group, and the Feigin-Frenkel conjecture on the simple critical characters can be thought of as a shadow of this connection. We discuss these topics and prove the first part of the Feigin-Frenkel conjecture, the restricted linkage principle.

Meinolf Geck (Aberdeen University)
Generic representations of finite groups of Lie type

This talk is concerned with modular representations of finite groups of Lie type in non-defining characteristic. We present a concise formulation of some major open problems and show how these are solved for principal series representations, where Hecke algebra methods can be used.

Viktor Ginzburg (University of Chicago)
Isospectral commuting variety and the Harish-Chandra D-module

Let $\g$ be a complex reductive Lie algebra with Cartan algebra $\h$. Hotta and Kashiwara defined a holonomic $D$-module $M$, on $\g\times\h$, called Harish-Chandra module. We give an explicit description of $gr M$, the associated graded module with respect to a canonical Hodge filtration on $M$. The description involves the isospectral commuting variety, a subvariety of $\g\times\g\times\h\times\h$ which is a finite extension of the variety of pairs of commuting elements of $\g$. Our main result establishes an isomorphism of $gr M$ with the structure sheaf of the normalization of isospectral commuting variety. It follows, thanks to Saito's theory of polarized Hodge modules, that the normalization of the isospectral commuting variety is Cohen-Macaulay and Gorenstein. This confirms a conjecture of M. Haiman.

In the special case where $\g=gl(n)$, there is an open subset of the isospectral commuting variety that is closely related to the Hilbert scheme of $n$ points in the plane. The sheaf $gr M$ gives rise to a locally free sheaf on the Hilbert scheme. We show that the corresponding vector bundle is isomorphic to the Procesi bundle. This was used by I. Gordon to deduce the positivity result for Macdonald polynomials, established earlier by Haiman.

Daniel Juteau (University of Caen)
Parity sheaves

(Joint work with Carl Mautner and Geordie Williamson)
Important problems in modular representation theory can be formlated in terms of perverse sheaves with positive characteristic coefficients, but the intersection complexes are very difficult to compute in this setting. With Carl Mautner and Geordie Williamson, we propose to study parity sheaves instead, beacuse they seem to be nicer and are also relevant to modular representation theory.

Masaharu Kaneda (Osaka City University)
Some observations on the structure of $F_*O_{G/P}$

We will present our hope that the Frobenius direct image $F_*O_{G/P}$ of the structure sheaf of $G/P$, $G$ a reductive algebraic group in positive characteristic and $P$ a parabolic subgroup, may admit indecomposable direct summands defined over $Z$, which describe by base change the category of coherent sheaves on the corresponding complex variety. This is joint work with Ye Jiachen.

Seok-Jing Kang (Seoul National University)
Crystal bases for quantum queer superalgebras

In this talk, we will explain how to develop crystal basis theory for quantum queer superalgebras $U_q(q(n))$. The notion of crystal bases will be modified and the tensor product rule will be presented. We will give an outline of proof for the existence of crystal bases.

* This is a joint work with D. Grantcharov, J. Jung, M. Kashiwara and M. Kim.

Toshiro Kuwabara (RIMS, Kyoto University)
Representation theory of the rational Cherednik algebra of type Z/lZ via micro local analysis

We will discuss the geometric representation theory for the rational Cherednik algebra of type Z/lZ. The rational Cherednik algebra of type Z/lZ is defined as a quantization of the Kleinian singularities of type A. This algebra is also called the finite W-algebra associated to subregular nilpotent elements in sl_l. Using the Deformation-Quantization algebras on a minimal resolution of the Kleinian singularities, we can establish an analogue of Beilinson-Bernstein correspondence for the rational Cherednik algebra. We will discuss an explicit construction of standard modules and irreducible modules in the category O through this Beilinson-Bernstein correspondence. As a consequence, we obtain regular holonomic systems from these modules. This talk is based on arXiv:1003.3407v1 [math.RT] and arXiv:1005.4645v1 [math.RT].

G. I. Lehrer (University of Sydney)
Equivariant K-theory of quantum homogeneous spaces and quantum affine spaces

I shall explain how to formulate and compute the equivariant K-theory of non-commutative spaces with quantum group symmetries. The results are in analogy with results of Bass and Haboush in the classical (commutative) case. This is joint work with R.B. Zhang.

Hyohe Miyachi (Nagoya University)
Hidden Hecke Algebras and Duality

I would like to talk about Hecke algebras which is hidden in $q$-Schur algebras. Those are Ext-algebras of certain semisimple modules. I would like to talk about a conjecture on the duality related to this Hecke algebras.

Kentaro Nagao (Nagoya University)
Donaldson-Thomas theory and cluster algebras

Non-commutative Donaldson-Thomas (ncDT) theory is the theory of moduli spaces of modules over a quiver with a potential. I will show that wall-crossing in ncDT theory provides a categorification of the theory of cluster algebras.

Hiraku Nakajima (RIMS, Kyoto University)
Cluster algebras and quantum affine algebras

I will explain what is a cluster algebra and why it is useful to understand the representation theory of a quantum affine algebra.

Tomoki Nakanishi (Nagoya University)
Dilogarithm identities in conformal field theory and cluster algebras

The dilogarithm identities for the central charges of conformal field theories were conjectured by Bazhanov, Kirillov , and Reshetikhin in the late 80's. They appeared naturally in the study of several integral models in the thermodynamic Bethe ansatz method. The functional generalizations of the identities were conjectured by Gliozzi-Tateo in the mid 90's. They are closely related to another conjecture on the periodicity of the Y-systems by Ravanini-Tateo- Valleriani. These dilogarithm conjectures have been partly proved by several authors with several methods. In this talk we present a proof of the conjectures in full generality based on the cluster algebra/ cluster category method recently developed by Fomin-Zelevinsky, Chapoton, Keller and others. The factorization property of the tropical Y-system is the key of the proof.

Masato Okado (Osaka University)
Stability in parabolic Lusztig q-analogues, one dimensional sums and fermionic formulas

There is a parabolic version of Lusztig's q-analogue of weight multiplicity. Lecouvey showed that in the large rank limit the parabolic Lusztig q-analogue of type B,C,D can be expressed as a sum of those of type A with Littlewood-Richardson coefficients. In this talk we report two other polynomials, called one dimensional sum and fermionic formula originating from the study of quantum integrable systems, also satisfy this equality, thereby proving these three polynomials are all equal. This talk is based on a joint work with Cedric Lecouvey and Mark Shimozono, and with Reiho Sakamoto.

Alexander Premet (University of Manchester)
On 1-dimensional representations of quantized Slodowy slices

In my talk I am going to discuss the latest results on 1-dimensional representations of finite W-algebras.

Daisuke Sagaki (Tsukuba University)
Tensor product multiplicities for crystal bases of extremal weight modules over quantum infinite rank affine algebras of types $B_{\infty}$, $C_{\infty}$, and $D_{\infty}$

We are going to discuss the crystal basis of an extremal weight module of integral extremal weight over the quantized universal enveloping algebra associated to the infinite rank affine Lie algebra of type $B_{\infty}$, $C_{\infty}$, or $D_{\infty}$. I am going to give an explicit description (in terms of Littlewood-Richardson coefficients) of how tensor products of these crystal bases decompose into connected components when their extremal weights are of nonnegative levels. These results, in types $B_{\infty}$, $C_{\infty}$, and $D_{\infty}$, extend the corresponding results due to Kwon, in types $A_{+\infty}$ and $A_{\infty}$; our results above also include, as a special case, the corresponding results (concerning crystal bases) due to Lecouvey, in types $B_{\infty}$, $C_{\infty}$, and $D_{\infty}$, where the extremal weights are of level zero.

This is a joint work with Satoshi Naito; this talk is based on arXiv:1003.2485.

Yoshihisa Saito (University of Tokyo)
Mirkovi\'c-Vilonen polytopes and quiver construction of crystal basis in type A

In this talk, we will give an explicit relationship between Mirkovi\'c-Vilonen polytopes and quiver construction of crystal basis in type A. As a by-product, we will give a new proof of the Anderson- Mirkovi\'c conjecture which describes the crystal structure of MV polytopes, by using language of quivers. We remark that it was already proved by Kamnitzer by combinatorial method.

Kentaro Wada (RIMS, Kyoto University)
On cyclotomic q-Schur algebras

The cyclotomic q-Schur algebra is one of the quasi-hereditary covers of the Ariki-Koike algebra, and it is defined as the endomorphism ring of a certain module of Ariki-Koike algebra. In this talk, after reviewing some structures of cyclotomic $q$-schur algebras, we give a presentation on cyclotomic q-Schur algebras by generators and fundamental relations using Cartan data of type gl.

Nanhua Xi (Chinese Academy of Sciences)
Some quotient algebras of affine Hecke algebras

An affine Hecke algebra has a big center. By modulo central characters one gets some finite dimensional quotient algebras. We will discuss a few particular quotient algebras. The information for the quotient algebras can be used to discuss isomorphisms between two affine Hecke algebras. Part of the work is a joint work with T. Shoji.

to Menu