Cohomology of bimultiplicative local systems on unipotent groups

Abstract

Let U1,U2 be connected commutative unipotent algebraic groups defined over an algebraically closed field k of characteristic p>0 and let L be a bimultiplicative Q‾ℓ-local system on U1×U2. In this paper we will study the Q‾ℓ-cohomology Hc⁎(U1×U2,L), which turns out to be supported in only one degree. We will construct a finite Heisenberg group Γ which naturally acts on Hc⁎(U1×U2,L) as an irreducible representation. We will give two explicit realizations of this cohomology and describe the relationship between these two realizations as a finite Fourier transform.