Dennis Gaitsgory

On the geometric Langlands conjecture

Let X be a (smooth, projective) curve and G be a reductive group over a finite field Fq. The field KX of rational functions on X is what number theorists call a global field and we can do the theory of automorphic functions over it. Namely, we consider the quotient G(KX)\G(A) (here A is the ring of adeles corresponding to KX) and study the space of functions on it, as a representation of the adele group G(A). It was essentially an observation of A.Weil that the quotient G(KX)\G(A) is closely related to the set of isomorphism classes of principal G-bundles on our curve X. However, G-bundles on X possess a richer structure: one can view the above set as a set of Fq-points of an algebraic variety (or, rather, a stack), denoted BunG. Moreover, instead of studying the vector space of functions on the “discrete” set of points of BunG, we will consider the category of `-adic sheaves on it. Basically, what people call the geometric Langlands program is an attempt to understand a spectral decomposition of the above category under the action of the so-called Hecke functors (the latter will be defined in the talk). The answer predicted by the Geometric Langlands Conjecture links this spectral decomposition to the moduli stack of local systems on X with respect to the Langlands dual group Ǧ.

Uhlenbeck Spaces via Affine Lie Algebras

Let G be an almost simple simply connected group over ℂ, and let Bun G a (ℙ2, ℙ1) be the moduli scheme of principalG-bundles on the projective plane ℙ2, of second Chern class a, trivialized along a line ℙ1 ⊂ ℙ2.

Geometric constant term functor(s)

We study the Eisenstein series and constant term functors in the framework of geometric theory of automorphic functions. Our main result says that for a parabolic

Weyl Modules and Opers without Monodromy

P\subset G

Parameters and duality for the metaplectic geometric Langlands theory

P⊂G with Levi quotient M, the !-constant term functor

Construction of central elements in the affine Hecke algebra via nearby cycles

\begin{aligned}{\text {CT}}_!:{\text {D-mod}}({\text {Bun}}_G)\rightarrow {\text {D-mod}}({\text {Bun}}_M)\end{aligned}