Lorentz Center - The Geometric Langlands Program from 7 Jul 2008 through 11 Jul 2008
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    The Geometric Langlands Program
    from 7 Jul 2008 through 11 Jul 2008

 
Program The Geometric Langlands 7-11 July 2008

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.

 

 

 

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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.

 

 

 

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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.

 

 

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Motivic Poisson summation

David Kazhdan

 

 


 

 

 


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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 $Bun_{G}$.

 

 

 

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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.

 

 

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An analytic version of the quantum geometric Langlands correspondence
Joerg Teschner


We will propose a variant of the so-called 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.

 

 


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Proof of the Kottwitz' conjecture on the transfer of Deligne-Lusztig

functions (joint with David Kazhdan)
Yakov Varshavsky (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).



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