A Workshop
on
Microlocal analysis on symplectic manifolds
In mathematics, functions on a manifold form a commutative algebra,
however in quantum physics,
they are not commutative but the differences of right and left products are power seires of $\hbar$. Such phenomenon is called deformation quantization.
In the present workshop, we shall have a basic course
by Masaki Kashiwara on the category of sheaves of deformation
quatization on a symplectic manifold with the Frobenius action
and its application to representation theory of rational Cherednik algebras. We shall have related lectures by Berard Leclerc, Toshiro Kuwabara and Kentaro Nagao.
Date: 
September 16  18, 2008 
Place: 
Seminar Room at IPMU Prefab. B, Kashiwa Campus of the University of Tokyo 
Organizer: 
Kyoji Saito(IPMU) 
Speaker: 
 Masaki Kashiwara (KURIMS, IPMU)
 Toshiro Kuwabara (KURIMS)
 Bernard Leclerc (Universite de Caen)
 Kentaro Nagao (Kyoto)

Schedule: 
 16 (Tuesday)
10:30  11:30   Kashiwara 
  lunch 
13:30  14:30   Leclerc 
  break 
15:00  16:00   Kuwabara 
 17 (Wednesday)
10:30  11:30   Kashiwara 
  lunch 
13:30  14:30   Leclerc 
  break 
15:00  16:00   Nagao 
 18 (Thursday)
10:30  11:30   Kashiwara 
  lunch 
13:30  14:30   Leclerc 
  break 

Title and abstract: 
 Masaki Kashiwara (KURIMS, IPMU)

"Quantization of symplectic manifolds"
The deformation quantization of a symplectic manifold
is a sheaf of rings on this symplectic manifold.
It contains a parameter $\hbar$, so that the category
of modules over this ring is a $\mathbd{C}[[\hbar]]$linear
category.
In the application, we need to eliminate $\hbar$ and
to obtain $\mathbf{C}$linear category.
In order to perform this, we propose the notion of Frobenius action.
I also explain its application
to the representation theory of rational Cherednik algebras.
 Toshiro Kuwabara (KURIMS)

"Characteristic cycles of standard modules for
the rational Cherednik algebra of type Z / l Z"
We study the representation theory of the rational Cherednik
algebra $H_\kappa = H_\kappa({\mathbb Z}_l)$ for
the cyclic group ${\mathbb Z}_l = {\mathbb Z} / l {\mathbb Z}$ and its connection with
the geometry of the quiver variety $\mathfrak{M}_\theta(\delta)$
of type $A_{l1}^{(1)}$.
We consider a functor between the categories of
$H_\kappa$modules with different parameters, called the
shift functor, and give the condition when it is an equivalence of
categories.
We also consider a functor from the category of
$H_\kappa$modules with good filtration to the category of
coherent sheaves on $\mathfrak{M}_\theta(\delta)$. We prove that
the image of the regular representation of $H_\kappa$ by
this functor is
the tautological bundle on $\mathfrak{M}_\theta(\delta)$. As a
corollary, we determine the characteristic cycles
of the standard modules.
It gives an affirmative
answer to a conjecture given
in \cite{Go} in the case of ${\mathbb Z}_l$.
 Bernard Leclerc (Universite de Caen)

"Nilpotent varieties and cluster algebras"
 Lusztig's lagrangian construction of the
enveloping algebra U(n) of the positive part n
of a KacMoody algebra g. This involves
constructible functions on the socalled
nilpotent varieties. This yields a basis
of U(n) called the semicanonical basis.
It is an open problem how this basis compares
with the canonical (or global) basis of the
quantum algebra U_q(n) specialized at q=1.
 For simplicity, assume that g is finitedimensional.
The graded dual of U(n) is isomorphic to
the coordinate ring C[N] where N is the unipotent
group with Lie(N) = n. Thus C[N] has a dual
semicanonical (and a dual canonical basis).
Fomin and Zelevinsky have introduced a cluster
algebra structure on C[N]. (I will recall what
this means). One of our main results is that the
cluster monomials of C[N] belong to the dual
semicanonical basis. They are naturally labelled
by the irreducible components of the nilpotent
varieties which have an open orbit under the group
of base change transformations.
The main idea is to remember that Lusztig's nilpotent
varieties are representation varieties of the
GelfandPonomarev preprojective algebra. So we can
use some machinery of the representation theory of
finitedimensional algebras.
 Generalization of (2) to the following cases:
 (a) N is replaced by the unipotent radical of a
standard parabolic subgroup of G.
 (b) N is replaced by the unipotent subgroup
N(w) = N \cap w^{1} N_{} w
where w belongs to the Weyl group and N_{} is
the opposite maximal unipotent.
In fact (b) makes sense in the KacMoody case
because N(w) is always finitedimensional.
 Kentaro Nagao (Kyoto)

"Flop invariance of curve counting on CalabiYau 3folds"
I will prove the flop invariance of euler numbers of DonaldsonThomas
and PandharipandeThomas moduli spaces on CalabiYau $3$folds. These
moduli spaces can be realized as moduli spaces of "perverse coherent
systems" at specific stability parameters. The flop invariance follows
as an application of Joyce's wallcrossing formula.
