Character Sheaves and Characters of Unipotent Groups over Finite Fields

Let G0 be a connected unipotent group over a finite field Fq, and let G = G0 ⊗Fq Fq, equipped with the Frobenius endomorphism Frq : G −→ G. For every character sheaf M on G such that Frq M ∼= M , we prove that M comes from an irreducible perverse sheaf M0 on G0 such that M0 is pure of weight 0 (as an `-adic complex) and for each integer n ≥ 1 the “trace of Frobenius” function tM0⊗FqFqn on G0(Fqn) takes values in Q , the abelian closure of Q. We further show that as M ranges over all Frq-invariant character sheaves on G, the functions tM0 form a basis for the space of all conjugation-invariant functions G0(Fq) −→ Q, 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 Frq-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 G0 is a possibly disconnected unipotent group over Fq. (In general, Frqinvariant character sheaves on G are related to the irreducible characters of the groups of Fq-points of all pure inner forms of G0 over Fq.)