Representation of Algebraic Groups and Quantum Groups

Representation Theory of Algebraic Groups and Quantum Groups

August 1 - 10, 2001

Sophia University, Tokyo


Henning H. Andersen (Aarhus University)
Small cells and fusion rules

Abstract: ( Tex image )
Let $G$ be a reductive algebraic group over a field of characteristic $p > 0$ and denote by $U_q$ the corresponding quantum group at a complex $p$-th root of unity $q$. Then tilting modules for $G$ can be "quantized" and give rise to tilting modules for $U_q$. In this talk we shall discuss the problem of decomposing these quantized tilting modules.

When the highest weight which parametrizes an indecomposable tilting module (for $G$ or for $U_q$) is in the lowest alcove then there is an explicit fusion rule which describes how many times this module occurs as summand in a given tilting module. Recent work by T. E. Rasmussen gives an algorithm for how to obtain corresponding information for weights in alcoves belonging to the second cell for the affine Weyl group associated with $G$. As an application one gets the dimensions of a class of simple modules for symmetric groups.

The same techniques can be used to compare tilting modules for $U_q$ to modules for the corresponding quantum group over a field $k$ of characteristic $l \neq p$. In this case one obtains among the applications equality of dimensions for certain simple modules for the Hecke algebras over $\mathbb{C}$ and $k$.

Susumu Ariki (Tokyo University of Mercantile Marine)
The representation type of Hecke algebras of type B

Abstract: ( Tex image ) (joint work with A. Mathas)
Hecke algebras of type $A$ and type $B$ are important objects in order to study the representations of finite classical groups of Lie type. Aiming at this application, Dipper and James started the modular representation theory of these Hecke algebras. Let ${\CH}_n$ be the Hecke algebra of type $B$ with two invertible parameters $q$ and $Q$ in a field $k$ such that its generators have quadratic relations $(T_0-Q)(T_0+1)=0$ and $(T_i-q)(T_i+1)=0$ $( 1 \le i < n )$. If $-Q$ is not a power of $q$, it has been proved in early '90s by combination of results of Uno and Dipper-James that ${\CH}_n$ is of finite representation type if and only if $n<2e$ where $q=\sqrt[e]{1}$. So, the remaining case is that $q=\sqrt[e]{1}$ and $Q=q^f$. By renormalizing $T_0$, we may assume that $0\le f\le e/2$. Assume that $e\ge 3$ for simplicity. Then our result is that ${\CH}_n$ is of finite representation type if and only if $ n < min (e,2f+4)$.

To prove that $n\ge min(e,2f+4)$ implies infinite representation type, we use a lemma saying that if an artinian algebra $A$ has an indecomposable projective module $P$ such that $End_A(P)$ is isomorphic to none of truncated polynomial rings $k[x]/(x^N)$, then it is of infinite representation type. To find such $P$, we need Dipper-James-Mathas' Specht module theory and my previous result on decomposition numbers, both proven for cyclotomic Hecke algebras of type $G(m,1,n)$.

To prove that $ n < min (e,2f+4)$ implies finite representation type, we use two array Maya diagrams to describe bipartitions. Then combinatorics of these Maya diagrams and James-Mathas' cyclotomic Jantzen-Schaper theorem prove the theorem.


Serguei Arkhipov (Independent University of Moscow)
Algebraic construction of quasi-Verma modules in positive characteristic

Abstract: ( Tex image )
In the talk we investigate a new class of infinite-dimensional modules over the hyperalgebra of a semi-simple algebraic group in positive characteristic $p$ called quasi-Verma modules. We present a purely algebraic construction of the global Grothendieck-Cousin complex corresponding to the standard line bumdle ${\mathcal L}(\lambda)$ on the Flag variety of the algebraic group stratified by the Schubert cells. We prove that the complex consists of direct sums of quasi-Verma modules for the highest weights of the form $w\cdot\lambda$ for various elements $w$ of the Weyl group. Finally we compare the obtained complex with the specialization of the quasi-BGG complex over the corresponding Lusztig quantum group into the base field of characteristic $p$.

Roman Bezrukavnikov (University of Chicago)
Localization type Theorems for modular representations and representations of quantum groups at a root of unity

Abstract: ( Tex image )
In a joint work with Mirkovic and Rumynin we show that a variant of Beilinson-Bernstein-Brylinski-Kashiwara localization theorem holds in (large) prime characteristic. There are indications that representations of the Kac-DeConcini quantum enveloping algebra at a root of unity admit a similar description; a close description of representations of the small quantum group is the subject of an ongoing joint work with Arkhipov and Ginzburg.

In the talk I will discuss these results and some applications (such as a description of the Grothendieck group of representations of a semi-simple Lie algebra in large prime characteristic with a fixed p-character in terms of coherent sheaves on the Springer fiber).

Michel Broue (Institut Henri Poincare)
Families of characters of cyclotomic Hecke algebras

Abstract: ( Tex image )
Some recent work by Gyoja and Rouquier have shown that Lusztig families of characters of Weyl groups may be viewed as blocks of the corresponding Hecke algebra on a suitable localisation of the ring of Laurent polynomials with one variable. This approach can be generalized in order to define families of characters of finite complex reflection groups - or, rather, of any cyclotomic Hecke algebra associated with a complex reflection group. These families are compatible with Lusztig families of unipotent characters of finite reductive groups as well as with all generalized Harish-Chandra partitions.

A preprint, which is the orign of this talk, can be obtained from here .

Charles W. Curtis (University of Oregon)
On the Gelfand-Graev representations of finite reductive groups: Zeta functions and functional equations

Abstract: ( Tex image )
Let $G$ be a finite reductive group, and let $\gamma$ be a Gelfand-Graev representation of $G$. For each complex valued function $\Phi$ on $G$ and each irreducible component $\pi$ of $\gamma$, a zeta function $Z(\Phi,\pi)$ is defined, and proved to satisfy a functional equation of the same form as the functional equation for zeta functions of irreducible cuspidal representations of $GL_{n}$ obtained by Springer and Macdonald (Math. Ann. 249, 1-15, 1980). Corresponding functional equations are obtained for the irreducible representations $f_{\pi}$ of the endomorphism algebra $H$ of $\gamma$. In case $\pi = \pm R_{T,\theta}$ the $\varepsilon$-factor in the functional equation is the Gauss sum of the representation $\pi$, as defined and computed by Kondo for the case of $GL_{n}$, and by Saito and Shinoda for a general finite reductive group. Applications to the representation theory of $GL_{n}$ are obtained. The talk is based on joint work with K. Shinoda.


Akihiko Gyoja (Nagoya University)
Prehomogeneous vector spaces and representations

Abstract: ( Tex image )
The $b$-functions of prehomogeneous vector spaces associated to nilpotent orbits of complex simple Lie algebras play important roles in the representation theory, which we shall discuss in this talk. We also give the explicit form of the $b$-functions. The latter is due to Y.Kaneko and myself (classical type) and to K.Ukai (exceptional type).

Jens C. Jantzen (Aarhus University)
Representations of Lie algebras in prime characteristics I
Representations of Lie algebras in prime characteristics II
Representations of Lie algebras in prime characteristics III

Abstract: ( Tex image )
Fix an algebraically closed field $K$ of characteristic $p > 0$. Consider a reductive algebraic group $G$ over $K$ and denote by $\frak g$ the Lie algebra of $G$. This series of lectures will be a survey on the representation theory of $\frak g$.

It is a particular feature of the theory in prime characteristics that all irreducible representations of $\frak g$ are finite dimensional. Therefore a classification of all irreducible representations of $\frak g$ appears to be feasible. In fact, such a classification does exist when $G$ is the general linear group. For arbitrary $G$, however, we have today (30 May 2001) still only partial information. And even in the case of GL$_n$ we do not know the dimensions of all irreducible representations.

Still, over the last six years considerable progress has been made in this area, beginning with Premet's proof of an old conjecture by Kac and Weisfeiler. In particular, there are in this area now some important conjectures by Lusztig that I want to discuss.

Masaharu Kaneda (Osaka City University)
On the Beilinson-Bernstein correspondence of D-modules on the flag variety in positive characteristic

Abstract: ( Tex image )
Let $G$ be a simply connected simple algebraic group, $B$ a Borel subgroup of $G$, $\cD^\lambda$ the sheaf of rings of the $\lambda$-twisted differential operators on the flag variety $X=G/B$, $\lambda$ a dominant weight, and $D^\lambda=\cD^\lambda(X)$.

In characteristic $0$ the Beilinson-Bernstein correspondence asserts that there is a natural surjection from the universal enveloping algebra of the Lie algebra of $G$ to $D^\lambda$ and that the flag variety $X$ is $D^\lambda$-affine.

In positive characterstic $\cD^\lambda$ admits, aside from the standard filtration, another filtration, called the $p$-filtration,constructed using the Frobenius morphisms on $X$, which Haastert described in terms of some induced representations from $B$ to the infinitesimal thickenings of $B$ and showed that any $\cD^\lambda$-module quasicoherent over $\cO_X$ is generated by the global sections even over $\cO_X$. We will make a progress report in positive characterstic.


Seok-Jin Kang (Seoul National University)
( Tex image )
1. Quantum groups and crystal bases

Basic facts on quantum groups and crystal bases. The definition and meaning of crystal bases. The existence and uniqueness of crystal bases. Tensor product rule for crystal graphs. Global bases.
2. Perfect crystals and Young walls
Theory of perfect crystals for quantum affine algebras. Path realization of crystal bases. Combinatorics of Young walls and crystal bases.
3. Fock space representation of quantum affine algebras
Representation of classical quantum affine algebras on the space of proper Young walls. LLT algorithm for global bases(=canonical bases) for classical quantum affine algebras.

Masaki Kashiwara (RIMS, Kyoto University)
On semisimple holonomic module conjecture

Abstract: ( Tex image )
I conjectured that semisimple holonomic $D$-modules are stable by projective direct images. Although I am still far from proving it, I will explain its consequences.

Norikai Kawanaka (Osaka University)
Sato-Welter games and Weyl groups of Kac-Moody Lie algebras

Abstract: ( Tex image )
Around 1950, Mikio Sato, partly inspired by papers of Nakayama on modular representations of symmetric groups, invented a 2-person game with a beautiful theory, but his work was never published in European languages. Almost at the same period, C.P. Welter independently created an equivalent theory, whose clear exposition is given in a famous book ("On Numbers and Games", Ch. 13, Academic Press, 1976) of J.H. Conway. In this talk, we give a vast generalization of the theory of Sato and Welter. This means that we now have a large collection of completely solvable 2-person games. The notion of miniscule elements of Weyl groups due to D. Peterson, and their combinatorial classification due to R.A. Proctor are essential in our study.

Anatol Kirillov (RIMS, Kyoto University)
Birational Representations of Symmetric Groups, Combinatorics and Discrete Integrable Equations

Abstract: ( Tex image )
We construct two birational representations of the extended affine symmetric group ${\widehat W}(A_{n+1}^{(1)})$ on the affine space $\mathbb{A}^{n^2}$, and show that {\it tropical version} of the Robinson-Schensted-Knuth correspondence is a birational automorphism of the affine space $\mathbb{A}^{n^2}$ which intertwines these two actions of the group ${\widehat W}(A_{n+1}^{(1)})$. Connections between tropical version of the Schutzenberger involution and certain discrete integrable equations will be explained.


Alexander Kleshchev (University of Oregon)
Projective representations of symmetric groups via Kac-Moody algebras

Abstract: ( Tex image )
We will consider modular representation theory of double covers of symmetric and alternating groups. The problems we are interested at are: classification of irreducible modules, their blocks and branching rules. We will explain how all of these can be obtained using certain affine algebras and the corresponding crystals.

Gustav I. Lehrer (University of Sydney)
Cell modules and standard modules for the affine Hecke algebra of type A

Abstract: ( Tex image )
Let $\wh H_n^a(q)$ be the Affine Hecke algebra of $G = GL_n$. Kazhdan-Lusztig and Ginzburg have defined "standard modules" $M_{s,N}^{\phi}$ for affine Hecke algebras in general, where $s \in G$ is semisimple, $N \in Lie(G)$ is nilpotent, $Ad(s)N = qN$ and $\phi$ is an irreducible representation of the group of the components of the simultaneous centralizer of $s$ and $N$. The standard modules generically (i.e. when $q$ is not a root of unity) have top quotients which form a complete set of irreducible representations of the algebra. When $q$ is a root of unity, not much is known about the composition factors of $M_{s,N}^{\phi}$.

My work with J. Graham on the affine Temperley-Lieb algebra defines, for $\wh H_a^n(q)$, certain "cell modules", and gives complete information on their composition series, even when $q$ is a root of unity. This is done using the diagram context of the algebras, and using "cellular structure". In this talk, I will explain how to identify cell modules with standard modules (in the Grothendieck ring). The link is via braid diagrams and the Temperley-Lieb algebra of $B_n$ (sometimes called the "blob algbera" by physicists).

George Lusztig (MIT)
Representations of graded Hecke algebras

Abstract: ( Tex image )
We will discuss the construction of irreducible representations of graded Hecke algebras using equivariant homology. A new ingredient is a proof of the induction theorem in this context which allows one to classify the square integrable representations.

Andrew Mathas (University of Sydney)
Tilting modules for cyclotomic Schur algebras

Abstract: ( Tex image )
The cyclotomic Schur algebras are endomorphism algebras of a direct sum of ``permutation like'' modules for the Ariki-Koike algebras: they include as special cases the q-Schur algebras of Dipper and James. These algebras were introduced partly to provide a new tool for studying the Ariki-Koike algebras and partly in the hope that they might generalize the beautiful Dipper-James theory which shows that the q-Schur algebras completely determine the modular representation theory of the $GL_n(q)$ in non-defining characteristic.

As yet there are no known (non type A) connections between the representation theory of the cyclotomic Schur algebras and that of the finite groups of Lie type; nonetheless the representation theory of these algebras is both rich and beautiful. For example, they are quasi-hereditary algebras and Jantzen's sum formula generalizes to this setting. In this talk I will survey the representation theory of the cyclotomic Schur algebras culminating with a description of their tilting modules.

Olivier Mathieu (Universite Claude Bernard Lyon)
Connections on stable bundles

Abstract: ( Tex image )
Let $X$ be a Riemmann surface, i.e a genus $g$ surface with a complex structure, and let $\Gamma$ be its fundamental group.

To each $n$-dimensional representation $\rho:\Gamma\rightarrow GL(n,{\bf C})$ corresponds a rank $n$ holomorphic bundle $E(\rho)$ on $X$. Its total space is $(\tilde X\times {\bf C}^{n})/\Gamma$, where $\tilde X$ is the universal cover of $X$, where $\Gamma$ acts diagonally on $\tilde X\times {\bf C}^{n}$ and where the action on the second component ${\bf C}^{n}$ is determined by $\rho$. Let $E$ be an holomorphic vector bundle on $X$. It is natural to ask: {\it Under which condition about $E$ is there a $\rho$ such that $E\simeq E(\rho)$?} Using the Riemann-Hilbert correspondance, this question is equivalent to: {\it Under which condition does $E$ admit an holomorphic connection?}.

Since any skew $(2,0)$ tensor on $X$ is zero, the curvature of any holomorphic connection vanishes. Therefore $c_{1}(E)=0$ whenever $E$ admits an holomorphic connection, where $c_{1}(E)$ is the first Chern class of $E$. The converse is almost true, as it has been proved by A. Weil: if $E$ is simple (i.e. any endomorphism is a scalar) and $c_{1}(E)=0$, then $E$ admits an holomorphic connection. However this holomorphic connection is not unique. In Weil's statement, the space of holomorphic connection is an affine space of dimension $1+(g-1)n^2$. The next question is: {\it Out of these connections, could we single out one nicer connection?}

This question has been answered by Narashiman and Seshadri {\bf [NS1][NS2]}. Their solution requires the notion of stability, which is now recalled. Indeed we will use the terminology "stable" instead of "stable of slope 0". An holomorphic vector bundle $E$ is called {\it stable} if $c_{1}(E)=0$, but $c_{1}(F)<0$ for any proper subbundle $F$. Here the meaning of the condition "$c_{1}(F)<0$" is explained by the natural identification $H^{2}(X)\simeq {\bf Z}$. Narashiman and Seshadri theorem states that a stable bundle $E$ admits a unique hermitian holomorphic connection, i.e. is isomorphic to $E(\rho)$ for a unique unitarisable representation $\rho$ (i.e. $\overline{\hbox{\rom{Im}}\,\rho}$ is a compact subgroup of $GL(n,{\bf C})$).

We will now use some new hypotheses. Let $K$ be a number field, let $X$ be a complete curve of genus $g$ and let $E$ be a stable vector bundle. The notion of stability is defined in this context, using the degree instead of the first Chern class. We can reformulate Narashiman and Seshadri theorem: for each $\sigma:K\rightarrow {\bf C}$, there is a unique hermitian connection on the bundle ${\bf C}\otimes E$ over $X\times_{K}{\bf C}$. Therefore to each infinite place of $K$ is attached a certain connection.

In {\bf [M]}, a similar statement for finite places of $K$ is proved. Also, a conjecture about the algebraicity of the solutions of certain differential equations is stated.

\bigskip {\it Bibiography:}

{\bf [M]}: O. Mathieu: Connections on stable bundles, preprint
{\bf [NS1]}: M.S. Narasimhan and C.S. Seshadri: Holomorphic vector bundles on a compact Riemann surface. Math. Ann. 155 (1964) 69-80.
{\bf [NS2]}: M.S. Narasimhan and C.S. Seshadri: Stable and unitary vector bundles on a compact Riemann surface. Ann. of Math. 82 (1965) 540-567.


Satoshi Naito (Tsukuba University)
Twining character formula for Demazure modules

Abstract: ( Tex image )
We talk about two kinds of approaches to a twining character formula for a Demazure module. Here a twining character is the graded trace of a Dynkin diagram automorphism $\omega$ of a module graded by its weights.

One approach is an algebro-geometric one (with M. Kaneda): we use the $\omega$-equivariant Demazure-Hansen desingularization of a Schubert variety.

Another approach is a combinatorial one (with D. Sagaki): we use the path model of a Demazure module and the global crystal base of a quantum Demazure module.

Hiraku Nakajima (Kyoto University)
Quiver varieties and quantum affine algebras

Abstract: ( Tex image )
Kazhdan-Lusztig and Ginzburg obtained a classification and character formulas of irreducible representations of an affine Hecke algebra using geometry of the Springer resolution of the nilpotent cone. I proved similar results for finite dimensional representations of a quantum affine algebra, using geometry of the so-called `quiver varieties'. In these talks, I will survey results related quiver varieties and quantum affine algebras.

Masatoshi Noumi (Kobe University)
Birational Weyl group actions, discrete integrable systems and tropical combinatorics

Abstract: ( Tex image )
I will give an overview of a general construction of birational Weyl group actions and its applications to discrete systems such as discrete KP, Toda and Painleve equations (based on the collaboration with K.Kajiwara and Y.Yamada). I will also discuss a remarkable relationship with totally positive matrices (Lusztig, Berenstein-Fomin-Zelevinsky) and tropical Robinson-Schensted-Knuth correspondence (Kirillov).

Viktor Ostrik (Independent University of Moscow)
Asymptotic Hecke algebra and central sheaves

Abstract: ( Tex image )
An asymptotic affine Hecke algebra $J$ was defined by G. Lusztig as a suitable limit of an affine Hecke algebra as parameter $q$ tends to 0. The knowledge of structure of $J$ gives a lot of information about the representation theory of the affine Hecke algebra for all values of $q$. Lusztig also proposed a Conjecture giving explicit and elementary description of algebra $J$ in terms of convolution algebras. This Conjecture was verified in many special cases by N. Xi. In this talk we explain geometric approach to Lusztig's Conjecture originated by R. Bezrukavnikov and based on the remarkable functor of central sheaves due to A. Beilinson, D. Gaitsgory and R. Kottwitz. This approach allows to prove statement which is somewhat weaker than original Lusztig's Conjecture but strong enough to reprove all previously known special cases, in particular the case of type $\tilde A_n$.


Kyoji Saito (RIMS, Kyoto University)
Elliptic Lie groups

Abstract: ( Tex image )
An (algebraic) elliptic group is introduced for an elliptic root system (= a root system belonging to a semi-positive lattice with the radical of rank 2). Since the Witt index of the lattice is larger than 1, the group is beyond the Kac-Moody groups. Its formal completion with respect to a marking admits a Bruhat-Iwahori-Matsumoto type decomposition. I conjecture the Chevalley type theorem on invariants for the elliptic groups.

Toshiaki Shoji (Science University of Tokyo)
Green functions associated to complex reflection groups

Abstract: ( Tex image )
Green dunctions of finite classical groups are determined by the data from Weyl groups and by certain combinatorial objects called symbols. Generalizing this, we define Green functions associated to complex reflection groups $G(e,p,n)$, and study their combinatorial properties.

In the case of $GL_n$, Green functions are obtained by modifying the Kostka polynomials, which are described as a transition matrix between two bases of symmetric functions, Hall-Littlewood functions and Schur functions. In our case, we define symbols associated to $G(e,p,n)$, and construct new type of symmetric functions, Schur functions and Hall-Littlewood functions parametrized by those symbols. As in the case of $GL_n$, it is shown that Green functions associated to $G(r,p,n)$ are obtained as a transition matrix between those two bases of symmetric functions. As a special case, our approach gives a combinatorial description of Green functions of type $B_n, C_n$ and $D_n$.

T.A. Springer (Utrecht University)
Intersection cohomology of large Schubert varieties

Abstract: ( Tex image )
Let $G$ be an adjoint semi-simple group and let $X$ be its wonderful compactification (introduced by De Concini and Procesi). Let $B$ be a Borel group of $G$. A large Schubert variety in $X$ is the closure of a double coset $BwB$ in $X$. In the talk I sketch a proof of the evenness of local and global intersection cohomology of large Schubert varieties and related varieties. A new kind of Kazhdan-Lusztig polynomials appears.

Toshiyuki Tanisaki (Hiroshima University)
Highest weight modules over affine Lie algebras

Abstract: ( Tex image )
I will talk about my joint work with M.\ Kashiwara concerning Kazhdan-Lusztig type character formulas for irreducible highest weight modules over affine Lie algebras. Especially, I will explain how $D$-modules and perverse sheaves on the infinite-dimensional flag manidols are related to modules over the affine Lie algebras.

Eric Vasserot (University of Cergy-Pontoise)
D-modules, formal loop spaces and chiral algebras

Abstract: ( Tex image )
To any scheme X of finite type we associate a formal loop space LX, which is an ind-scheme of ind-infinite type. If X is smooth, the De Rham complex of a right D-module on LX has a natural structure of vertex algebra. We prove that the direct image, on X, of this complex is the Chiral De Rham complex introduced by Malikov, Schechtman, Vaintrob.

Nanhua Xi (Chinese Academy of Sciences)
A property of distinguished involutions in an affine Weyl group of type $\tilde A_n$ (tentative)

Abstract: ( Tex image )
We shall show that a conjecture of Lusztig on the distinguished involutions of affine Weyl groups is true for type $\tilde A_n$.

Andrei Zelevinsky (Northeastern University)
An introduction to cluster algebras

Abstract: ( Tex image )
We discuss a new class of commutative algebras introduced in a joint work with Sergey Fomin. This is an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups.