Tempered local geometric Langlands
by Sam Raskin
Abstract: The (arithmetic) Langlands program is a cornerstone of modern representation theory and number theory. It has two incarnations: local and global. The former conjectures the existence of certain "local terms," and the latter predicts remarkable interactions between these local terms. By necessity, the global story is predicated on the local.
Geometric Langlands attempts to find similar patterns in the geometry of curves. However, the scope of the subject has been limited by a meager local theory, which has not been adequately developed.
The subject of this talk is a part of a larger investigation into local geometric Langlands. We will give an elementary overview of the expectations of this theory, discuss a certain concrete conjecture in the area (on "temperedness"), and provide evidence for this conjecture. One application of our results is a proof of Beilinson-Bernstein localization for the affine Grassmannian for GL_2, which was previously conjectured by Frenkel-Gaitsgory.