 
Summary: KOSZUL DUALITY AND SEMISIMPLICITY OF FROBENIUS
PRAMOD N. ACHAR AND SIMON RICHE
Abstract. A fundamental result of BeilinsonGinzburgSoergel states that
on flag varieties and related spaces, a certain modified version of the category
of adic perverse sheaves exhibits a phenomenon known as Koszul duality.
The modification essentially consists of discarding objects whose stalks carry
a nonsemisimple action of Frobenius. In this paper, we prove that a number
of common sheaf functors (various pullbacks and pushforwards) induce cor
responding functors on the modified category or its triangulated analogue. In
particular, we show that these functors preserve semisimplicity of the Frobe
nius action.
1. Introduction
Let X be a variety over a finite field Fq. In Deligne's work on the Weil conjec
tures [D1, D2], a central role is played by the category of "mixed constructible com
plexes of Q sheaves" on X, denoted DWeil
(X) in the present paper. (Henceforth,
we will avoid calling this category "mixed," as that conflicts with the terminology
of [BGS].) In order to belong to DWeil
(X), a complex F must have the property
that the eigenvalues of the Frobenius action on stalks of F at Fqn points of X are of
