Character sheaves and characters of unipotent groups over finite fields

Let $G_0$ be a connected unipotent group over a finite field ${\Bbb F}_q$, and let $G=G_0\otimes_{{\Bbb F}_q}\overline{\Bbb F}_q$, equipped with the Frobenius endomorphism ${\rm Fr}_q:G\rightarrow G$. For every character sheaf $M$ on $G$ such that ${\rm Fr}_q^*M\cong M$, 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 $\ell$-adic complex) and for each integer $n\geq 1$ the ``trace of Frobenius'' function $t_{M_0\otimes_{\Bbb F}_q}}{\Bbb F}_{q^n}}$ on $G_0({\Bbb F}_{q^n})$ takes values in ${\Bbb Q}^{ab}$, the abelian closure of $\Bbb{Q}$. We further show that as $M$ ranges over all ${\rm Fr}_q^*$-invariant character sheaves on $G$, the functions $t_{M_0}$ form a basis for the space of all conjugation-invariant functions $G_0({\Bbb F}_q)\rightarrow {\Bbb Q}^{ab}$, and are orthonormal with respect to the standard {\it 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 ${\rm Fr}_q^*$-invariant $\Bbb{L}$-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 ${\Bbb F}_q$. (In general, ${\rm Fr}_q^*$-invariant character sheaves on $G$ are related to the irreducible characters of the groups of ${\Bbb F}_q$-points of all pure inner forms of $G_0$ over ${\Bbb F}_q$.)