Abstracts
Let $G$ be a reductive algebraic group over an
algebraically closed field $k$ of characterstic $p>0$ with
${\mathfrak g}=\mathrm{Lie}(G)$.
I will talk about my result with D. Nakano concerning the following
problem.
Given a nilpotent orbit ${\mathcal O}$ in the restricted nullcone
${\mathcal N}_{1}({\mathfrak g})$, construct
a finite-dimensional (tilting) $G$-module
such that the support variety of $M$, ${\mathcal V}_{\mathfrak g}(M)$,
is the closure of ${\mathcal O}$.
(with M.Kashiwara and M.Okado)
Theory of geometric crystals, which is introduced by
A.Berenstein and D.Kazhdan, is a geometrical analogue
of Kashiwara's crystal theory.
First, we review the theory of geometric crystals and
unipotent crystals in Kac-Moody setting.
We also introduce the notion of
tropicalizations(Trop)/ultra-discretizations(UD).
Next, we construct some geometric crystals associated with affine
Kac-Moody algebras. And we see that they
are tropicalizations of certain perfect crystals.
Finally, tropical R for the affine geometric crystals are
given explicitly.
The localization theorem of D-modules by Beilinson and Bernstein
has found fundamental application in the representation theory. We will discuss a relation of tilting
sheaves on a smooth projective variety and the localization of D-modules in positive characteristic.
This is a joint work with Hashimoto Yoshitake and Dmitriy Rumynin.
It is known that, in every extremal even unimodular lattice,
the set of vectors of a given norm forms a spherical design of certain
strength. This means that, for example, the root system of type $E_8$
can be used to approximate the integrals of continuous functions
defined on the sphere of radius $\sqrt{2}$ quite accurately, in the
sense that it is exact up to polynomial approximation of degree
less than or equal to 7. In this talk, we give, under some conditions,
the converse of this statement holds. That is, certain spherical
designs necessarily come from an extremal even unimodular lattice.
The examples include, in addtion to the root system $E_8$,
the set of shortest vectors of the Leech lattice. The results
for these two examples were obtained by Bannai and Sloane in 1981.
Our approach is much simpler, and applicable to dimensions up to 72.
This is joint work with Boris Venkov.
In this talk, I plan to give a survey on the work related to the theory
of spherical designs. An emphasis will be on the current status of the
overall theory of spherical designs, rather than the explanation of a single
result. I will discuss some (but perhaps not all) of the following topics.
(1) Concept of spherical design and examples.
(2) Classification problems of tight spherical designs.
(3) Connection with the kissing number problems and sphere packing problems.
(4) Spherical designs which are orbits of finite groups.
(5) Spherical designs attached to Euclidean lattices, in particular to Type II
extremal lattices. (Venkov's theorem and connection to modular forms.)
(6) Generalizations of the concept of spherical designs.
(6-1) to various projective spaces (rank one symmetric spaces) and
Grassmannian spaces.
(6-2) to Euclidean spaces (Euclidean designs and Gaussian designs. )
(7) Connection with the theory of orthogonal polynomials of several variables.
In 1964 N. Iwahori published a paper containing a presentation with
generators and relations of the Hecke algebra $H(G,B)$ of a finite
Chevalley group $G$ with respect to a Borel subgroup $B$. The Hecke
algebra is isomorphic to the centralizer algebra of the permutation
representation of
$G$ on the cosets of $B$ over the field of complex numbers. The
presentation obtained by Iwahori showed that the Hecke algebra can be
viewed as a deformation of the group algebra of the Weyl group $W$ of
$G$, and led to connections between the representation theory of $W$
and the representation theory of $G$. These connections have proved to
be important in later work on the representation theory of $G$, and are
surveyed in the talk.
Another Hecke algebra, isomorphic to the centralizer algebra ${\cal H}$
of a Gelfand-Graev representation
of a finite Chevalley group $G$, was investigated by T. Yokonuma in a
paper published in 1967.
Yokonuma proved that the Hecke algebra was commutative, so that the
Gelfand-Graev representation associated with it is multiplicity free.
The talk will include further discussion of Yokonuma's theorem on the
commutativity of ${\cal H}$ along with remarks on subsequent work on
the computation of the irreducible representations of the Hecke algebra
${\cal H}$ and their connection with the virtual representations
$R_{T,\theta}$ of Deligne and Lusztig.
We discuss a new setting of extended affine
Lie algebras, and the so-called Kac-conjecture.
We also create and study the associated groups
in case of nullity 2.
* Robert V. Moody (U. Alberta)
Diffraction and dynamical systems in aperiodically ordered structures
Diffraction has been a main stay in crystallography for nearly a
century. It has also played a key role in the theory of quasicrystals.
The discovery of these "non-periodic crystals" has generated a lot of
new ideas as mathematicians and physicists have tried to getter a deeper
understanding of the phenemon of long-range aperiodic order.
One of fruitul ideas coming out of statistical mechanics and ergodic
theory has been the use of dynamical systems and their spectral theory.
Here rather than a single disordered structure, one groups together a
whole family of them, which can be related either by time evolution of
the system, or as we will see it, by the translational symmetries of space.
In this talk we will discuss aperiodic order in the context of
diffraction, and dynamical systems and show the intimate relationship
that connects them. We will also introduce some ideas that go beyond
the case of just translational (Abelian) symmetries to the non-Abelian
world, and indicate some results of Prof. Takeo Yokonuma in this
wider context.
* T. Yokonuma (Sophia U.)
Discrete sets and associated dynamical systems in a non-commutative setting
We define a uniform structure on the set of discrete sets of a locally
compact topological space on which a locally compact topological group
acts continuously. Then we investigate the completeness of these
uniform spaces.
* Y. Gomi (Sophia U.)
The Markov traces and the Fourier transforms
Let $\mathcal H$ be an Iwahori-Hecke algebra and $W$ the corresponding
Coxeter group.
If $\mathcal H$ is of type $A$, $B$ or $D$, then we have the Markov
trace $\tau_z$ on $\mathcal H$.
When we specialize $z$ to $0$, the $\tau_z$ specializes to $\tau_0$,
the canonical symmetrizing trace on $\mathcal H$.
Since the weights of $\tau_0$ are written by using the generic degrees
of $\mathcal H$, we can consider the weights of $\tau_z$ as analogue
of the generic degrees.
We know that the generic degrees are obtained from the fake degrees of
$W$ by applying the so-called Fourier transform which is constructed
by Lusztig.
The fake degree of $W$ obtained from the Molien series of $S(V)$,
the symmetric algebra of $V$ where $V$ is the vector space which
affords the reflection representation of $W$.
As analogue of the fake degrees,
we consider the Molien series of $S(V) \otimes L(V)$ where $L(V)$ is
the exterior algebra of $V$.
Then we determine a trace function $\tau$ on $\mathcal H$ whose
weights are given by applying the Fourier transform to the above
Molien series.
When $\mathcal H$ is of type $A$, the $\tau$ coincides with the Markov
trace and when $\mathcal H$ is of type $B$ or $D$, the $\tau$
coincides with some specialized Markov trace.
For any finite type, the $\tau$ satisfies some certain properties
which allow us to call it the Markov trace on $\mathcal H$.
* T. Shoji (Nagoya U.)
Symmetric space associated to finite special linear groups
(Joint work with K. Sorlin)
Let $G$ be a connected reductive group defined over a finite
field $F_q$ with Frobenius map $F$. The $G^{F^2}$-module
$X_G = G^{F^2}/G^F$ is regarded as an analogue of the symmetric
space in the real case, and is called a finite symmetric space
assocaited to $G$. For each irreducible character $\roh$ of
$G^{F^2}$, we denote by $m(\roh)$ the multiplicity of $\roh$ in
$X_G$. In the case where $G = GL_n$ with standard Frobenius map,
Gow showed that $X_G$ is multiplicity free, and determined
$m(\rho)$ for all irreducible characters $\rho$. Kawanaka
determined $m(\rho)$ for almost all $\rho$ in the case where
$G$ is reductive with connected center, and is simple modulo
center, by using an interesting connection between finite
symmetric spaces and Delinge-Lusztig's virtual characters. Then
Lusztig gave a general formula for $m(\rho)$ under the same setting,
by pushing forward the connection with Delinge-Lusztig theory.
In this talk, we take up the case where $G^{F} = SL_n(F_q)$, which
is the first example that the center of $G$ is disconnected.
We determine $m(\rho)$ for all irreducible characters, up to some
minor ambiguity. In particular, we can show that
$m(\rho) \in \{ 0,1,2\}$. We also discuss the relationship between
finite symmetric spaces and the theory of Shintani descent.