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The Geometric Langlands Program |
Abstracts of speakers (in alphabetical order): Mirabolic Deligne-Langlands correspondence Victor Ginzburg We introduce `mirabolic analogues' of the
affine flag variety and of the affine Grassmannian, certain
interesting varieties that have arisen in the works on Cherednik
algebras of type A. We introduce
a `mirabolic analogue' of the affine Hecke algebra and discuss a conjectural classification of its finite
dimensional representations analogous to the one given by Deligne-Langlands-
Lustig in the case of the affine Hecke
algebra. -------------------- Towards
Langlands Duality for Conjugacy
Classes Serge Gukov Abstract: In this talk, based on a joint work with E.Witten,
we will seek a bijection between conjugacy classes in a
complex group G and in the Langlands dual group ^LG.
Motivated by the ideas from physics, we present a list
of invariants which in many cases allow to identify dual pairs of conjugacy
classes, including Lusztig's special unipotent classes as well as new examples of duality between rigid
semi-simple conjugacy classes. -------------------- Quiver
varieties and double affine Grassmannian Hiraku Nakajima Abstract : Braverman and Finkelberg recently propose an affine analog of the
geometric Satake correspondence via moduli spaces of $G$-instantons
on a simple singularity of type $A_l$. These moduli spaces play the role of affine Grassmanian
for an affine Lie algebra for $G$ (so double affine Grassmannian). When $G$ is of type $A_r$, the moduli spaces
are quiver varieties, and are related also to an affine Lie algebra, but is of type $A_l$ by my
work some years ago. Two affine Lie algebras are related by I.Frenkel's
level-rank duality. We then discuss the convolution diagrams for tensor products and branching are interchanged in the two
picture. -------------------- Motivic Poisson summation David Kazhdan ------------------------- On
Arthur SL2 in the Geometric Langlands correspondence Vincent Lafforgue Let $G$ be a reductive group over $C$ and $X$ a smooth projective curve
over $C$. Then $H^{*}(Bun_{G})$ acts on the derived category
of $D$-modules on $Bun_{G}$, and therfore should
act on the derived category of local systems on $X$ for the dual group. We give a guess for this action and check
it in the case of tori. Then we try to understand what object in the derived category
of local systems on $X$ for the dual group corresponds to the constant $D$-module on $Bun_{G}$. -------------------- -------------------- Shift
of argument algebras Leonid Rybnikov (joint work with B. Feigin and E. Frenkel) The universal enveloping algebra of any simple Lie algebra g contains a
family of commutative subalgebras (parametrized by regular elements of g), called the quantum shift of argument subalgebras.
We prove that generically their joint pectra on
finite- dimensional modules are in bijection
with the set of monodromy- free opers
for the Langlands dual group of G
on the projective line with regular singularity at one point and irregular singularity of order two at another point.
In addition, we show that the highest weight vector
in any irreducible finite-dimensional g-module is a cyclic vector for
the quantum shift of argument subalgebra
corresponding to a regular nilpotent element of g. This gives the structure of a graded Frobenius
algebra on any such module. -------------------- An
analytic version of the quantum geometric Langlands
correspondence
motivated from mathematical physics based on
representations of the affine Kac-Moody algebra with noncritical
level. An important new ingredient of our proposal is the (partially established) existence of a scalar product on the
relevant spaces of conformal blocks. This version of the quantum geometric Langlands correspondence thereby gets a more analytic flavor which may strengthen some analogies
to the versions of the Langlands correspondence involving spaces of automorphic forms. A partly conjectural connection to the quantization of the (higher) Teichmueller spaces (Kashaev; Fock, Goncharov...) will also be discussed.
Proof
of the Kottwitz' conjecture on the transfer of Deligne-Lusztig functions (joint with David Kazhdan) The transfer conjecture of Langlands-Shelstad
(which is now a theorem due to Waldspurger and Ngo) asserts that for every function $f$
on a p-adic group $G$ there is a function $f^H$ on its endoscopic group $H$ such that $f$ and $f^H$
have "matching orbital integrals". The famous "fundamental lemma", which was conjectured by Langlands and recently completed by Ngo, describes function $f^H$ in the case when $G$ is unramified
and $f$ is bi-invariant under a hyperspecial
subgroup of $G$. Though for a general function $f$ one does not expect to construct $f^H$ explicitly, it would be desirable to have a greater supply of
functions $f$ for which $f^H$ is known. In mid 90'th Kottwitz
gave a conjectural description of $f^H$ in the case
when $f$ is an inflation of the character of a Deligne-Lusztig
representation. In our work with Kazhdan we prove a
generalization of this conjecture (under some mild restriction on the residue characteristic). [Back] |