On Euler characteristic of equivariant sheaves

Abstract

Let k be an algebraically closed field of characteristic p>0 and let ℓ be another prime number. Gabber and Looser proved that for any algebraic torus T over k and any perverse ℓ-adic sheaf F on T the Euler characteristic χ( F) is non-negative. We conjecture that the same result holds for any perverse sheaf F on a reductive group G over k which is equivariant with respect to the adjoint action. We prove the conjecture when F is obtained by Goresky–MacPherson extension from the set of regular semi-simple elements in G. From this we deduce that the conjecture holds for G of semi-simple rank 1.