Cohomology of bimultiplicative local systems on unipotent groups

Let $U_1, U_2$ be connected commutative unipotent algebraic groups defined over an algebraically closed field $k$ of characteristic $p>0$ and let $\mathcal{L}$ be a bimultiplicative $\overline{\mathbb{Q}}_\ell$-local system on $U_1\times U_2$. In this paper we will study the $\overline{\mathbb{Q}}_\ell$-cohomology $H^*_c(U_1\times U_2,\mathcal{L})$, which turns out to be supported in only one degree. We will construct a finite Heisenberg group $\Gamma$ which naturally acts on $H^*_c(U_1\times U_2,\mathcal{L})$ 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.