  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.
