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Event

Lecture Event
SFB-Seminartag
Prof. Kurke, Prof. Dancer, Prof. Grigoryan
17 Jul 2007, 15:00 – 19:00

Program

Talk
On the geometric Langlands Conjecture
Prof. Dr. Herbert Kurke
17 Jul 2007, 15:00 – 16:00

Abstract: Due to the very limited time I have to pretend that everybody in the audience knows things like "algebraic stacks" (which intuitively is something like an algebraic variety where every point carries some extra-information), "derived categories", "theory of ${\cal D}$-modules" (a slight generalization of "holomorphic vectorbundle endowed with a flat meromorphic connection") etc. (Some more details I will give in a talk in the seminare "Algebraic Geometry" at FU, Monday, July 16, 4 pm - 6 pm in Arnimallee 3, SR 119.)


First we explain Langland-duality for reductive groups G, G', this means: The lattice X(T') is dual to the lattice X(T) (T, T' maximal tori in G, G', $X(T)={\rm Hom}(T,\C^*)$) such that the rootsystem $R'\subset X(T')$ of G' is the corootsystem of G (and vice versa). E.g. SLn and PSLn are Langlands dual, SO(2n+1) and Sp(2n) are Langlands dual, whereas SO(2n) or GLn are both Langlands-dual to itself:


Now, let X be a compact Riemann surface. We have various geometric objects associated with it, for example, for each reductive group G:

  1. the algebraic stack of holomorphic G-principal bundles ${\rm Bun}(X,G)$;
  2. the algebraic stack of local systems ${\rm Loc}(X,G)$ (G-bundles with a flat holomorphic connection, analytically it is the space ${\rm Hom}(\pi_1(X),G)/$conjugation).
Both are smooth stacks, and standard techniques, e.g. theory of coherent sheaves, theory of ${\cal D}$-modules, and its derived categories are available.
The GLC is the following: One expects an equivalence between compactly supported derived categories (of bounded complexes) of coherent sheaves on ${\rm Loc}(X,G)$ and of coherent ${\cal D}$-modules on ${\rm Bun}(X,G')$. Moreover, this correspondence should associate to points [E] of ${\rm Loc}(X,G)$ (identified with the 1-dimensional skyscraper-sheaf supported in [E]) so-called "automorphic ${\cal D}$-modules" = "Hecke-Eigensheaves to E". We will explain this notion in some detail, and results obtained in this direction.

If time permits we also will explain how GLC is related to the "classical" Langlands conjecture for global fields (= theorem of Lafforgue, for function fields over finite fields).


Lecture Events 2007