# The motivic Satake equivalence

NUMBER THEORY

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April 7, 2021 3:00 pm - 4:00 pm
via Zoom Video Conferencing
##### Speaker:

The geometric Satake equivalence due to Lusztig, Drinfeld, Ginzburg, Mirković and Vilonen is an indispensable tool in the Langlands program. Versions of this equivalence are known for different cohomology theories such as Betti cohomology or algebraic D-modules over characteristic zero fields and $\ell$-adic cohomology over arbitrary fields. In this talk, I explain how to apply the theory of motivic complexes as developed by Voevodsky, Ayoub, Cisinski-Déglise and many others to the construction of a motivic Satake equivalence. Under suitable cycle class maps, it recovers the classical equivalence. As dual group, one obtains a certain extension of the Langlands dual group by a one dimensional torus. A key step in the proof is the construction of intersection motives on affine Grassmannians. A direct consequence of their existence is an unconditional construction of IC-Chow groups of moduli stacks of shtukas. My hope is to obtain on the long run independence-of-$\ell$ results in the work of V. Lafforgue on the Langlands correspondence for function fields. This is ongoing joint work with Jakob Scholbach from Münster.