CHARACTER SHEAVES AND CHARACTERS OF UNIPOTENT GROUPS OVER FINITE FIELDS

Let G(0) be a connected unipotent group over a finite field F-q, and let G = G(0) circle times(Fq) (F) over bar (q), equipped with the Frobenius endomorphism Fr-q : G -> G. For every character sheaf M on G such that Fr-q* M congruent to M we prove that M comes from an irreducible perverse sheaf M-0 on G(0) such that M-0 is pure of weight 0 (as an l-adic complex) and for each integer n >= 1 the "trace of Frobenius" function t(M0 circle times Fq) F-qn on G(0)(F-qn) takes values in Q(ab), the abelian closure of Q. We further show that as M ranges over all Fr-q*-invariant character sheaves on G, the functions t(M0) form a basis for the space of all conjugation-invariant functions G(0)(F-q) -> Q(ab), and are orthonormal with respect to the standard unnormalized Hermitian inner product on this space. The matrix relating this basis to the basis formed by the irreducible characters is block-diagonal, with blocks corresponding to the Fr-q*-invariant L-packets (of characters or, equivalently, of character sheaves). We also formulate and prove a suitable generalization of this result to the case where G(0) is a possibly disconnected unipotent group over F-q. (In general, Fr-q*-invariant character sheaves on G are related to the irreducible characters of the groups of F-q-points of all pure inner forms of G(0) over F-q.)