Quadratic $\ell$-adic sheaf and its Heisenberg group

In this paper, we introduce a new class of $\ell$-adic sheaves, which we call quadratic $\ell$-adic sheaves, on connected unipotent commutative algebraic groups over finite fields. They are sheaf-theoretic enhancements of quadratic forms on finite abelian groups in the spirit of the function-sheaf dictionary. We show that a certain finite Heisenberg group acts on a quadratic sheaf and that the cohomology of the quadratic sheaf gives an irreducible representation of the group. We also compute the Frobenius eigenvalues of the cohomology groups. As a byproduct, we find a large number of examples of affine supersingular varieties.