Journal of the Institute of Mathematics of Jussieu

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Journal of the Institute of Mathematics of Jussieu (2010), 9:719-739 Cambridge University Press
Copyright © Cambridge University Press 2010

Research Article

Twisted geometric Satake equivalence

Michael Finkelberga1 and Sergey Lysenkoa2

a1 Independent Moscow University, Institute for Information Transmission Problems and State University Higher School of Economy, Department of Mathematics, 20 Myasnitskaya Street, Moscow 101000, Russia (
a2 Institut Élie Cartan Nancy (Mathématiques), Université Henri Poincaré Nancy 1, BP 70239, 54506 Vandoeuvre-lés-Nancy Cedex, France (
Article author query
finkelberg m [Google Scholar]
lysenko s [Google Scholar]


Let k be an algebraically closed field and O = k[[t]] ⊂ F = k((t)). For an almost simple algebraic group G we classify central extensions 1 → $\mathbb{G}_m\to E\to G(\bm{F})\$mEG(F) → 1; any such extension splits canonically over G(O). Fix a positive integer N and a primitive character ζ : μN(K) →$\mathbb{Q}}_\ell^*}$ (under some assumption on the characteristic of k). Consider the category of G(O)-bi-invariant perverse sheaves on E with $\mathbb{G}_m$m-monodromy ζ. We show that this is a tensor category, which is tensor equivalent to the category of representations of a reductive group ǦE,N. We compute the root datum of ǦE,N.

(Received September 25 2008)

(Accepted October 14 2008)

Key Wordsgeometric Langlands program; Satake isomorphism; monodromic perverse sheaves

AMS 2010 Mathematics subject classificationPrimary 14D24; Secondary 22E57; 11R39