 
Summary: CONSTRUCTIBLE SHEAVES ON AFFINE GRASSMANNIANS
AND GEOMETRY OF THE DUAL NILPOTENT CONE
PRAMOD N. ACHAR AND SIMON RICHE
Abstract. In this paper we study the derived category of sheaves on the affine
Grassmannian of a complex reductive group G, contructible with respect to the
stratification by G(C[[x]])orbits. Following ideas of Ginzburg and Arkhipov
BezrukavnikovGinzburg, we describe this category (and a mixed version) in
terms of coherent sheaves on the nilpotent cone of the Langlands dual re
ductive group G. We also show, in the mixed case, that restriction to the
nilpotent cone of a Levi subgroup corresponds to hyperbolic localization on
affine Grassmannians.
1. Introduction
1.1. Let G be a complex connected reductive group, and let
Gr G := G(K)/ G(O)
be the associated affine Grassmannian (where K = C((x)) and O = C[[x]]). The
Satake equivalence is an equivalence of tensor categories
SG : Rep(G)
 P G(O)eq(Gr G)
between the category P G(O)eq(Gr G) of G(O)equivariant perverse sheaves on Gr G
