Γ-sheaves on Reductive Groups

Let G be a reductive group over a finite field F = Fq. Fix a non-trivial additive character ψ : F → Q × l . In [3] we introduced certain γ-functions γG,ρ,ψ on the set Irr(G) of irreducible representations of the finite group G = G(F ). As usual every function γG,ρ,ψ on Irr(G) gives rise to an AdG-equivariant function ΦG,ρ,ψ on G. The purpose of this paper is to construct an irreducible perverse sheaf ΦG,ρ,ψ on G such that the function ΦG,ρ,ψ is obtained conjecturally by taking traces of Frobenius morphism in the stalks of ΦG,ρ,ψ. In order to do this we need to assume that ρ satisfies certain technical condition (we call ρ good if that condition is satisfied). We prove this conjecture for G = GL(n) and for G of semi-simple rank one. We also prove the above conjecture assuming that certain cohomology vanishing for the sheaf ΦG,ρ,ψ holds (we show that this is the case for G of semisimple rank 1). Assuming this vanishing we show that if both ρ1 and ρ2 are good then ΦG,ρ1,ψ ⋆ ΦG,ρ2,ψ ≃ ΦG,ρ1⊕ρ2,ψ. We also compute the convolution of ΦG,ρ,ψ with the majority of Lusztig’s character sheaves. We conjecture that the functor of convolution with ΦG,ρ,ψ is exact in the perverse t-structure.