The motivic Satake equivalence

Mardi 3 décembre 2019 14:00-15:15 - Timo Richarz - Technische Universität Darmstadt

Résumé : The geometric Satake equivalence due to Lusztig, Drinfeld, Ginzburg, Mirković and Vilonen is by now an indispensable tool in the Langlands program. Versions of this equivalence for different cohomology theories are known such as Betti cohomology or algebraic D-modules over characteristic zero fields and l-adic cohomology over arbitrary fields. In this talk I explain how to apply the theory of étale motives developed by Voevodsky, Ayoub, Cisinski-Déglise and many others to the construction of a motivic Satake equivalence which under suitable realization functors recovers the geometric Satake equivalence. As dual group one obtains a certain extension of the classical Langlands dual group by a one dimensional torus. This relates to the notion of C-algebraic versus L-algebraic introduced by Buzzard and Gee. 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. Our hope is to obtain on the long run independence-of-l results in the work of V. Lafforgue on the Langlands correspondence for global function fields. This is ongoing joint work with J. Scholbach from Münster.

Lieu : salle 3L15 bâtiment 307

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