of Chevalley Groups and Related Topics

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* Abstract: *

Kac-Wakimoto admissible representations are (conjecturally all) modular invariant representations of affine Kac-Moody algebras, which are also important for (affine) W-algebras. In this talk we will prove two fundamental conjectures on Kac-Wakimoto admissible representations in full generality: One is the Feigin-Frenkel conjecture on the singular support of G-integrable admissible representations. The other is the Adamovic-Milas conjecture on the rationality of the associated vertex operator algebras in the category O.

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** Cédric Bonnafé **
(Montpellier)
** Calogero-Moser cells: the smooth case **

* Abstract: *
(Joint work with R. Rouquier)
Using the geometry of Calogero-Moser spaces (i.e. centre
of Cherednik algebras at t=0), we have defined a notion
of Calogero-Moser cells for complex reflection groups.
We conjecture these cells to coincide with Kazhdan-Lusztig
cells: in this talk, we will show that, whenever the
Calogero-Moser space is smooth, then the Calogero-Moser
cells have the sizes predicted by our conjecture.

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** John Enyang **
(Sydney)
** Cellular bases for algebras with a Jones basic construction **

* Abstract: *

We define a method which produces explicit cellular bases for algebras obtained via a Jones basic construction. For the class of algebras in question, our method gives formulae for generic Murphy--type cellular bases indexed by paths on branching diagrams and compatible with restriction and induction on cell modules. Our construction allows for a uniform combinatorial treatment of cellular bases and representations of the Brauer, Birman--Murakami--Wenzl, Temperley--Lieb, and partition algebras, among others.

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** Meinolf Geck **
(Aberdeen)
** Kazhdan-Lusztig cells and the Frobenius-Schur indicator **

* Abstract: *

Let W be a finite Coxeter group. It is well-known that the number
of involutions in W is equal to the sum of the degrees of the
irreducible characters of W. Following a suggestion of Lusztig,
we show that this equality is compatible with the decomposition
of W into Kazhdan-Lusztig cells. The proof uses a generalisation
of the Frobenius-Schur indicator to symmetric algebras, which
may be of independent interest. Along the way, we establish some
properties of left cell modules which previously were only known
to hold in the equal parameter case;

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** Osamu Iyama **
(Nagoya)
** Pseudo-tilting theory **

* Abstract: *

The notion of tilting modules was introduced by Brenner-Butler. As a shadow of the structure of the derived category, the set of tilting modules has an interesting combinatorics, e.g. tilting mutation due to Riedtmann-Schofield and Happel-Unger is closely related to cluster theory. In this talk, we introduce a notion of pseudo-tilting modules as a generalization of tilting modules. We develop a theory of pseudo-tilting mutation, which completes tilting mutation and has a cluster structure. This is a joint work with Idun Reiten.

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** Masaharu Kaneda **
(Osaka City University)
** On the structure of parabolically induced $G_1T$-Verma modules **

* Abstract: *
Let $\cP_\bbC$ be a complex homogeneous projective variety.
Write
$\cP_\bbC=G_\bbC/P_\bbC$
with a complex reductive group
$G_\bbC$ and a parabolic subgroup
$P_\bbC$ of $G_\bbC$.
These groups are defined over
$\bbZ$ and have counterparts
$G$ and $P$ in positive characteristic.
Let $G_1$ be the kernel of the Frobenius endomorphism of $G$
and
$T$ a maximal torus of
$P$.
Ye Jiachen and I have found for $G$ of low rank or in case $G/P$ is a projective space a recipe to construct
a complete strongly exceptional sequence of coherent sheaves on $\cP_\bbC$ from
parabolically induced $G_1T$-Verma modules.
I will report on the current status of study on the structure of parabolically induced $G_1T$-Verma modules.

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** Shu Kato **
(Kyoto University)
** A homological study of Green polynomials **

* Abstract: *

Kostka polynomials are certain family of polynomials indexed by two copies of simple modules of a Weyl group $W$. They are intimately connected with unipotent characters in the sense of Deligne-Lusztig, and they admits a characterization by the Lusztig-Shoji algorithm.
In 2001, Shoji further defined Kostka polynomials for some complex reflection groups by utilizing $\ell$-symbols and the (abstraction of the) above characterization.
In this talk, we reinterpret Shoji's definition of Kostka polynomials in terms of homological algebra. This naturally upgrades Kostka polynomials to a family of indecomposable modules that we call Kostka systems. They give a new characterization of Kostka polynomials in the case $W$ is a Weyl group. As applications, we see that:

a) every total homology group of a Springer fiber is generated by its top-term;

b) there exists a transition formula between Kostka polynomials of type B/C;

c) as long as we consider the ordinary representations, two standard modules of affine Hecke algebra have no extension (at all degree) if they are ordered correctly.

If time/research allows, we discuss several other applications related to Kostka systems. (The items a)--c) might be replaced by other things in case the research goes faster than expected.)

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** Noriaki Kawanaka **
(Kwansei Gakuin University)
** Algorithms and games with hook structure **

* Abstract: *

Since the seminal paper of T. Nakayama on modular properties of representations of a symmetric group, the importance of the notion of hooks of a Young diagram has been well-recognized. The purpose of this talk is to present an axiomatic theory of hooks, which enables us
to look at old problems from a new and more general point of view.

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** Gus I. Lehrer **
(Univ. of Sydney)
** The Brauer category and second fundamental theorem of invariant theory **

* Abstract: *

Let $V$ be the natural module for the orthogonal or
symplectic group $G$ over a field of
characteristic zero.Brauer proved in 1937 that there is a surjective
homomorphism from the Brauer
algebra $B_r(x)$ to $\End_G(V^{\otimes r})$. I shall show how to prove
that the kernel of this homomorphism
is generated by a single idempotent element of the Brauer algebra,
even in the generic case when the algebra
is not semisimple. A-pplications to the quantum case and positive
characteristic will be discussed.
This is joint work with R. Zhang.

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** Frank Lübeck **
(RWTH Aachen)
** On problems with computing character values of finite groups of Lie type **

* Abstract: *

In this talk I want to come back to an old project whose ultimate goal would be to develop a computer program that gets as input the root datum of a connected reductive algebraic group together with an automorphism of finite order. These data define for each prime power $q$ a finite group of Lie type $G(q)$. The output of the program should be a parameterization of the ordinary character tables of all these groups $G(q)$ (so called "generic character tables").
This talk will contain a quick overview of what can be done by automatic programs so far, and then mention some of the problems which remain to achieve the ultimate goal in full generality.

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** George Lusztig **
(MIT)
** A bar operator for involutions in a Coxeter group **

* Abstract: *
Let M be the Q(u) vector space spanned by the involutions in a Coxeter group W. In this lecture I will define a Hecke algebra structure and a compatible bar operator on M, using which one can define polynomials P^\sigma_{y,,w} for any two involutions in W, which give a refinement of the polynomials P_{y,w} defined in my 1979 paper with Kazhdan. This extends an earlier joint result with Vogan which dealt with the case where W is a Weyl group.

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** Jean Michel **
(Paris VII)
** Parabolic Deligne-Lusztig varieties **

* Abstract: *

The geometric version of the Broué conjectures for the principal block of finite reductive groups predicts that the endomorphism algebra of the cohomology of certain Deligne-Lusztig varieties is a cyclotomic Hecke algebra. The varieties are associated to roots of the Garside element in a Garside category. This talk will report progress by the author and F. Digne, and independently by O. Dudas in understanding these varieties.

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** Tomoki Nakanishi **
(Nagoya)
** Classical and quantum dilogarithm identities **

* Abstract: *
The pentagon identity of the dilogarithm has been known for a long time since Abel, and it is related to various subjects in mathematics. Recently it was recognized that it is associated with the periodicity of the Y-system of type $A_2$, or equivalently, the periodicity of the cluster algebra of type $A_2$. Similarly, the pentagon identity of the quantum dilogarithm by Kashaev and Faddeev is associated with the periodicity of the quantum cluster algebra of type $A_2$. In this talk I will present the generalization of the pentagon identities of the classical and quantum dilogarithm associated with the periods of mutations of classical and quantum cluster algebras.

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** Hiroyuki Ochiai **
(Kyushu)
** Double flag variety for a symmetric pair and finiteness of orbits **

* Abstract: *

Let G be a complex reductive algebraic group and K the fixed point
subgroup of an involutive automorphism of G. Consider the product of (
partial) flag varieties of G and K with the diagonal action of K, which
is called a `double flag variety'. We study a double flag variety of
finite type, i.e., the case where the number of K-orbits are finite. We
mention a connection with the spherical action. We give several examples
of double flag varieties of finite type, and for type AIII, we give a
classification of double flag varieties of finite type in the case where
the flag variety of G is of full type. This is a joint work with Kyo
Nishiyama (Aoyama Gakuin).

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** Soichi Okada **
(Nagoya)
** Spinor representations and symmetric functions **

* Abstract: *

Symmetric functions are useful in the representation theory of classical groups. In this talk, we introduce a family of symmetric functions with coefficients in the ring of integers adjoining a new element e with the property e^2 = 1, and investigate their properties. These symmetric functions can be used to describe the irreducible decompositions of tensor products and restrictions involving spinor representations of the pin groups.

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** Nobuharu Sawada **
(Sophia University)
** TBA **

* Abstract: *

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** Ken-ichi Shinoda**
(Sophia University)
** Gauss sums on finite groups and Hecke algebras **

* Abstract: *

Reviewing properties of Gauss sums on finite groups,
we introduce sums on Hecke algebras of type A, after Y. Gomi,
extending properties of Gauss sums on symmetric groups.

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** Karine Sorlin**
(Amiens)
** Modular Springer Correspondence for classical types **

* Abstract: *

In 1976, T. A. Springer defined a correspondence making a link between the irreducible ordinary (characteristic zero) representations of a Weyl group and the geometry of the associated nilpotent variety. This correspondence plays a main role in the theory of ordinary representations of finite groups of Lie type. In 2007, D. Juteau defined a modular Springer correspondence (in positive characteristic). In a recent work in common with D. Juteau and C. Lecouvey, we have determined the modular Springer correspondence for classical types.

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** Toshiyuki Tanisaki **
(Osaka City University)
** TBA **

* Abstract: *

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** Kentaro Wada **
(Shinshu)
** Drinfeld type realization of cyclotomic q-Schur algebras **

* Abstract: *

The cyclotomic q-Schur algebra is a quasi-hereditary cover of an Ariki-Koike algebra. In this talk, we give (countable infinite) generators of the cyclotomic q-Schur algebra, and describe the relations for such generators (not all defining relations of the cyclotomic q-Schur algebra). These relations are very similar to the relations of Yangians (or quantum loop algebras). In the time writing this abstract, I have not developed any applications from this loop-type of arguments, which are known for Yangians (quantum loop algebras). However, I believe that this description (perhaps, under some modification) is important, and have many applications.

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