Workshop

The geometric Satake equivalence is an equivalence between representations of the dual group and equivariant perverse sheaves on the affine Grassmannian. This can be viewed as a local statement happening over a fixed point on a global curve. In this talk I will explain a version of the geometric Satake equivalence over a power of a global curve. I will also describe how this construction is compatible with certain operations over the Beilinson-Drinfeld Grassmannians, such as convolution and fusion.

Vincent Lafforgue's work implies the existence of the "automorphic-to-Galois" direction of the Langlands correspondence for reductive groups G over a global field of positive characteristic. I will discuss a potential converse to this for Galois representations with Zariski dense image in the dual group \hat{G}. This is joint work with Gebhard Bockle, Michael Harris, and Chandrashekhar Khare.

The l-adic pro-semisimple completion of a profinite group packages its l-adic representations valued in (not necessarily connected) semisimple groups. When the profinite group considered is the étale fundamental group of a smooth variety over a finite field k, Drinfeld has proven an "independence-of-l" result for these pro-semisimple completions. This talk will describe the result precisely, and explain how Drinfeld ultimately reduces it to the existence of compatible systems containing a given l-adic local system on X. This existence result in turn follows from the Langlands correspondence when X is a curve, and in general from an earlier result of Drinfeld reducing the general case to the case of curves.