Representation of Algebraic Groups and Quantum Groups 06

Representation Theory
of Algebraic Groups and Quantum Groups 06

June 12 - 17, 2006

Noyori Conference Hall, Nagoya Univesity, Nagoya


Henning H. Andersen (Aarhus University)
Some quotient categories of modular representations

Abstract: ( Tex image )
Let $G$ be a reductive algebraic group over a field of characteristic $p>0$ and consider the category of finite dimensional $G$-modules. Relati\ ve to some choice of positive roots for $G$ we fix a dominant weight $\lambd\ a$. Then we shall introduce a quotient category consisting of modules whose composition factors are "close to $\lambda$" (we shall make the meaning o\ f this precise in the talk). Our aim is now to study the effect of adding a $p$-multiple of a weight to $\lambda$. We show that under appropriate conditions this leaves the quotient category in question invariant. As a consequence we derive some invariance properties of composition factor multiplicities in Weyl modules with highest weights close to $\lambda$. We get a similar invariance property for the Weyl factor multiplicities i\ n the corresponding indecomposable tilting modules. Our arguments also apply to quantum groups at roots of unity.

Tomoyuki Arakawa (Nara Women's Univesity)
Representations of W-algebras

Abstract: ( Tex image )
Abstract: (Affine) W-algebras are an interesting family of vertex algebras, which can be viewed as chiralization of the Kostant-Lynch theory. In this talk, we will describe their representation theory using the method of quantum reduction.

Susumu Ariki (RIMS, Kyoto University)
Non-recursive characterization of Kleshchev bipartitions

Abstract: ( Tex image )
We give a purely combinatorial and non-recursive criterion for when Specht module $S^{(\lambda,\mu)}$ of Hecke algebra of type B has non-zero quotient $D^{(\lambda,\mu)}$. The proof uses Littelmann's path model, in particular. This is a joint work with Kreiman and Tsuchioka.


Jie Du (Univesity of New South Wales)
Linear quivers, Generic extensions and Kashiwara operators

Abstract: ( Tex image )
One of the key properties of the quantum group associated to a semisimple complex Lie algebra is the existence of its canonical basis defined independently by Lusztig and Kashiwara. This basis induces bases in all finite highest weight modules over the corresponding Lie algebra. The properties of the canonical basis are closely tied up in the properties of the associated combinatorics, arising from the various parametrization approaches. In this talk, I will give an explicit comparison between the parametrizations via Kashiwara operators and via generic extensions for quiver representations.

The existence of generic extensions for a Dynkin or cyclic quiver $Q$ gives rise to a graph $\sf G$, called the generic extension graph. This graph has the same vertex set $\Lambda_Q$ as the crystal graph $\sf C$ for the quantized enveloping algebra associated to $Q$. If $P(X)$ denotes the set of all paths between $0$ and $\lambda\in\Lambda_Q$ in graph $X$, we may define two maps $$\wp_Q:P({\sf G})\to\Lambda_Q\qquad\text{and} \qquad\kappa_Q:P({\sf C})\to\Lambda_Q,$$ by sending a path to the endpoint $\lambda$. The two path sets can be naturally identified as the set $\Omega$ of all words on the alphabet $I$, the vertex set of $Q$. Thus, we obtain maps $\wp_Q,\kappa_Q:\Omega\to\Lambda_Q$. We prove that $\kappa_Q(w)\le\wp_Q(w)$ for all $w\in\Omega$ and $Q$ under the Bruhat type ordering on $\Lambda_Q$. Moreover, if $Q$ is a (finite or infinite) linear quiver, then the intersection of the fibres $\wp_Q^{-1}(\lambda)$ and $\kappa_Q^{-1}(\lambda)$ is non-empty for every $\lambda\in\Lambda_Q$. We will also show that this non-emptyness property fails for a cyclic quiver.

This is a joint work with Bangming Deng and Guanglian Zhang

Meinolf Geck (Aberdeen University)
Kazhdan-Lusztig cells and cellular bases

Abstract: ( Tex image )
Let $H$ be an Iwahori--Hecke algebra associated with a finite Weyl group. In type $A_n$, it is known that the Kazhdan--Lusztig basis of $H$ is cellular in the sense of Graham--Lehrer. We report on a new, elementary proof of this result. Our methods also provide, for the first time, purely algebraic proofs of fundamental properties of Lusztig's $a$-function (in type $A_n$). Similar arguments apply to type $B_n$, for a certain class of unequal parameters.

Victor Ginzburg (Univesity of Chicago)
Symplectic reflection algebras and Hilbert schemes

Abstract: ( Tex image )
We discuss symplectic reflection algebras from the point of view of holomorphic symplectic geometry. We then discuss in more detail the case of Cherednik algebras associated to an algebraic curve and corresponding analogues of character sheaves.


Jens C. Jantzen (Aarhus University)
Representations of the Witt-Jacobson algebras in prime characteristic

Abstract: ( Tex image )

Seok-Jing Kang (Seoul National University)
Crystal bases for quantum generalized Kac-Moody algebras

Abstract: ( Tex image )
Abstract: We develop the crystal basis theory for quantum generalized Kac-Moody algebras. We define the notion of crystal bases for $U_q({^_Nathfrak g})$-modules in the category ${^_Nathcal O}_{int}$ and prove the standard properties of crystal bases including the tensor product rule. We then prove that there exist crystal bases (and global bases) for $V(^_Mambda)$ $(^_Mambda ^_Jn P^{+})$ and $U_q^{-}({^_Nathfrak g})$. We also introduce the notion of abstract crystals for quantum generalized Kac-Moody algebras and study their fundamental properties. Finally, we prove the crystal embedding theorem and give a characterization of the crystals $B(^_Jnfty)$ and $B(^_Mambda)$.

Masaki Kashiwara (RIMS, Kyoto University)
Equivariant K-theory of affine flag manifolds

Abstract: ( Tex image )
We consider the equivariant K-thoery of affine flag manifolds with respect to the Borel action. The Scubert varities, the closure of Borel orbits, are finite-codimensional subvarieties of the flag manifolds, and they are normal and Cohen-Macaulay. We represent the class of the structure sheaf of the Scubert varieties by Laurent polynomials.

Syu Kato (University of Tokyo)
An exotic Deligne-Langlands correspondence for symplectic groups

Abstract: ( Tex image )
The Deligne-Langlands-Lusztig conjecture (proved by Kazhdan- Lusztig, Ginzburg) asserts that each simple module of an affine Hecke algebra corresponds to some geometric datum. An affine Hecke algebra of type $C$ admits a natural two-parameter deformation $\mathbb H $ (and this is best possible in some sense).

In this talk, we realize $\mathbb H$ as the equivariant $K$- group of a certain variety, which we refer as the exotic Steinberg variety. This enables us to present a Deligne-Langlands type classification of simple $\mathbb H$-modules when the values and ratios of deformation parameters are not too bad.

G. I. Lehrer (University of Sydney)
Endomorphism algebras of tensor powers

Abstract: ( Tex image )
Let $\mathfrak g$ be a complex semisimple Lie algerba and $U_q$ the Drinfel'd-Jimbo quantisation over the field ${\mathbb C}(q)$ of rational functions in the indeterminate $q$. If $V$ and $V_q$ are corresponding irreducible modules for $\mathfrak g$ and $U_q$, it is known that in several cases, $\text{End}_{U_q}V_q^{\otimes r}$ is a deformation of $\text{End}_{\mathfrak g}V^{\otimes r}$, and both algebras have a cellular structure, which in principle permits one to study non-semisimple deformation of either.

We present a framework (\lq\lq strongly multiplicity free'' modules) where the endomorphism algebras are \lq\lq generic'' in the sense that in the classical (unquantised) case, they are quotients of Kohno's infintesimal braid algebra $T_r$, while in the quantum case, they are quotients of the group ring ${\mathbb C}(q)B_r$ of the $r$-string braid group $B_r$. In addition to the well known cases above, these include the irreducible 7 dimensional module in type $G_2$ and arbitrary irreducibles for $\mathfrak s\mathfrak l_2$.

This is joint work with Ruibin Zhang


George Lusztig (MIT)
( Tex image )
Graded Lie algebras and intersection cohomology

Let $i$ be a homomorphism of the multiplicative group into a connected reductive algebraic group over $C$. Let $G^i$ be the centralizer of the image $i$. Let $LG$ be the Lie algebra of $G$ and let $L_nG$ ($n$ integer) be the summands in the direct sum decomposition of $LG$ determined by $i$. Assume that $n$ is not zero. For any $G^i$-orbit $O$ in $L_nG$ and any irreducible $G^i$-equivariant local system $L$ on $O$ we consider the restriction of some cohomology sheaf of the intersection cohomology complex of the closure of $O$ with coefficients in $L$ to another orbit $O'$ contained in the closure of $O$. For any irreducible $G^i$-equivariant local system $L'$ on $O'$ we would like to compute the multiplicity of $L'$ in that restriction. We present an algorithm which helps in computing that multiplicity.

Ian Mirkovic (University of Massachusetts)
Exotic hearts for triangulated categories of coherent sheaves on cotangent bundles of flag varieties

Abstract: ( Tex image )
This is a joint work with Bezrukavnikov and Rumynin. It is a step in Bezrukavnikov's program of verification of Lusztig's elaborate predictions of the numerical structure of representation theory for Lie algebras in positive characteristic. For a semisimple algebraic group in positive characteristic a version of the Beilinson-Bernstein formalism provides localization of Lie algebra representations to D-modules on the flag variety. In this setting, the analogue of the ``Riemann-Hilbert'' correspondence leads not to perverse sheaves but to twisted coherent sheaves on the cotangent bundle to the flag variety. This provides an exotic t-structure on coherent sheaves that corresponds to the standard one on representations. The fact that this t-structure is (roughly), defined over integers, is a categorical formulation of the independence of the representation theory on the characteristic.

Hyohe Miyachi (Nagoya University)
Runner Removal Morita Equivalence

Abstract: ( Tex image )
Joint work with Joseph Chuang

In my talk I would like to talk about representation theory of symmetric groups, Iwahori-Hecke algebras of type A and their $q$-Schur algebras at roots of unity.

The focus here is to compare module categories of Hecke algebras in different roots of unity. The result even works for any blocks at any roots of unity in postive characteristics with some suitable assumptions and has some applications for James' conjecture which is consistent with Lusztig's character formula conjecture for type A.

Our method for the new results is based on the knowledge of on so-called Rouquier block algebras and derived equivalences of blocks algebras in Hecke algebras in terms of Chuang-Rouquier's sl2-categorifications and perverse Morita equivalences.

Hiraku Nakajima (Kyoto University)
$q$--characters and crystal bases of finite dimensional representations of quantum affine algebras

Abstract: ( Tex image )
We consider irreducible finite dimensional representations of quantum affine algebras. In 2000 I computed their $q$-characters, but the computation involved a combinatorial algorithm similar to the definition of Kazhdan-Lusztig polynomials. Thus the answers were not explicit. In this talk I will give some examples of explicit formulas and also some examples of crystal bases.

Toshiki Nakashima (Sophia University)
Tropical R for affine geometric crystals

Abstract: ( Tex image )
We shall introduce several tropical R's for affine geometric crystals explicitly. We can perform it by using certain involution on M-matrix of simply-laced affine Lie algebras.

Raphael Rouquier (University of Leeds)
Deligne-Lusztig varieties and modular representations

Abstract: ( Tex image )
Broue has conjectured relations between Deligne-Lusztig varieties and modular representations of finite groups of Lie type in non-describing characteristic. I will discuss joint work with Bonnafe, where we follow Lusztig's approach for characteristic zero, and have to deal with new geometrical problems.


Daisuke Sagaki (Tsukuba University)
Lakshmibai-Seshadri paths of level-zero weight shape and level-zero representations of quantum affine algebras

Abstract: ( Tex image )
Let $U_q(\mathfrak{g})$ be a quantum affine algebra with weight lattice $P$, and let $\lambda = \sum_{i \in I_{0}} m_i \varpi_{i} \in P$, with $m_{i} \in \mathbb{Z}_{\geq 0}$, be a level-zero dominant integral weight, where $\varpi_{i} \in P$, $i \in I_{0}$, are the level-zero fundamental weights. In this talk, we first give an explicit description of the crystal structure of the $P$-weighted crystal $\mathbb{B}(\lambda)$ of all Lakshmibai-Seshadri paths (LS paths) of shape $\lambda$, and then explain its relation to the crystal base of the extremal weight module of extremal weight $\lambda \in P$.

Second, we give a tensor product decomposition theorem for the $P_{cl}$-weighted crystal $\mathbb{B}(\lambda)_{cl}$, that $\mathbb{B}(\lambda)_{cl} \simeq \bigotimes_{i \in I_0} \mathbb{B}(\varpi)_{cl}^{\otimes m_{i}}$, and then explain its relation to the crystal base of the corresponding tensor product of level-zero fundamental representations $W(\varpi_{i})$, $i \in I_{0}$, of a quantum affine algebra $U_{q}^{\prime}(\mathfrak{g})$ with weight lattice $P_{cl}$.

Finally, we give an interpretation via LS paths of one-dimensional configuration sums associated to the level-zero fundamental representations $W(\varpi_{i})$, $i \in I_{0}$. In particular, in the case of type $A_{\ell}^{(1)}$, this gives a description of the Kostka-Foulkes polynomials in terms of LS paths.

This is a joint work with Satoshi Naito

Yoshihisa Saito (University of Tokyo)
On Hecke algebras associated with elliptic root systems

Abstract: ( Tex image )
In the study of singularity theory, Kyoji Saito introduced a generalization of the theory of root systems (called elliptic root systems) and their Weyl groups (called elliptic Weyl groups). An elliptic Weyl group is the group generated by reflections attached to all roots in an elliptic root system. It is not Coxeter group, but its generators and relations are described by elliptic Dynkin diagrams due to Kyoji Saito.

In this talk we will talk on a q-analogue of elliptic Weyl groups called elliptic Hecke algebras. Firstly we will give a definition of elliptic Hecke algebra attaching to elliptic Dynkin diagrams. After that, we will discus on the relationship between double affine Hecke algebras and elliptic Hecke algebras.

Olivier Schiffmann (ENS)
Elliptic Hall algebra and double affine Hecke algebras

Abstract: ( Tex image )
We give a realization of the (spherical) double affine hecke algebras of type A as a convolution algebra of functions (or perverse sheaves) on the moduli spaces of coherent sheaves on elliptic curves. This provides an interpretation of Macdonald polynomials (in type A) in terms of certain Hecke eigensheaves.


Takeshi Suzuki (RIMS, Kyoto University)
Conformal Field Theory and Double Affine Hecke Algebras

Abstract: ( Tex image )
Abstract: We study about highest weight representations of the double affine Hecke algebra (DAHA, or Cherednik algebra) of type $A$ and their relation with representations of the affine Lie algebra based on the conformal field theory (CFT) called Wess-Zumino-Witten model.

We will see that the CFT on the complex plane ${\mathbb{C}}$ gives a natural and explicit construction of a representation of the rational DAHA as a certain space of coinvariants constructed from highest weight representations of the affine Lie algebra.

Similarly, the CFT on the cylinder ${\mathbb{C}}\setminus\{0\}$ gives a construction of representations of the trigonometric DAHA (the degenerate DAHA). Our space of coinvariants coincides, after localization, with the vector bundle given by the CFT, and the construction above gives a functor between categories of highest weight representations of the affine Lie algebra and DAHA which is compatible with the `Knizhnik-Zamolodchikov functor' between categories of representations of DAHA and the (affine) Iwahori-Hecke algebra studied by Ginzburg-Guay-Opdam-Rouquier and Varagnolo-Vasserot.

When a parameter (called the level) of the theory is generic, the functor from the plane model gives the classical Schur-Weyl reciprocity between the finite-dimensional Lie algebra and the symmetric group, and the functor from the cylinder model is reduced to the functor from the Bernstein-Gelfand-Gelfand category of the finite-dimensional Lie algebra to the category of finite-dimensional representations of the degenerate affine Hecke algebra. This functor has been introduced and studied by Arakawa-Suzuki-Tsuchiya.

We focus on integrable representations of the affine Lie algebra when the level is a positive integer. It turns out that irreducible integrable representations correspond through the functor to a special class of irreducible representations of DAHA which have a combinatorial description by `periodic' standard tableaux and have a nice character formula described by level-restricted Kostka polynomials.

Toshiyuki Tanisaki (Osaka City University)
Kazhdan-Lusztig basis and a geometric filtration of an affine Hecke algebra

Abstract: ( Tex image )
I will talk about my joint work with N. Xi (math.RT/0411304). According to Kazhdan-Lusztig and Ginzburg, the Hecke algebra of an affine Weyl group is identified with the equivariant $K$-group of Steinberg's triple variety. The $K$-group is equipped with a filtration indexed by closed $G$-stable subvarieties of the nilpotent variety, where $G$ is the corresponding reductive algebraic group over $\mathbb{C}$. It is conjectured that the filtration is compatible with the Kazhdan-Lusztig basis of the Hecke algebra. In this talk we will give its proof in the case of type $A$.


Jie Xiao (Tsinghua University)
Derived categories and Lie algebras

Abstract: ( Tex image )
The talk is based on a joint work with F.Xu and G.Zhang. It consists of :
(1) the moduli spaces of derived categories;
(2) constructible functions and the convolution multiplication;
(3) octahedral axiom and Jacobi identity;
(4) realization of infinite dimensional Lie algebras.