of Algebraic Groups and Quantum Groups '10

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* Abstract: *

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 **

* Abstract: *

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 **

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

(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 $F_*O_{G/P}$
**

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

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 $D_{\infty}$
**

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

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
**

* Abstract: *

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.

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