Paper detail

Character sheaves and characters of unipotent groups over finite fields

Let G_0 be a connected unipotent algebraic group over a finite field F_q, and let G be the unipotent group over an algebraic closure F of F_q obtained from G_0 by extension of scalars. If M is a Frobenius-invariant character sheaf on G, we show that M comes from an irreducible perverse sheaf M_0 on G_0, which is pure of weight 0. As M ranges over all Frobenius-invariant character sheaves on G, the functions defined by the corresponding perverse sheaves M_0 form a basis of the space of conjugation-invariant functions on the finite group G_0(F_q), which is orthonormal with respect to the standard unnormalized Hermitian inner product. The matrix relating this basis to the basis formed by irreducible characters of G_0(F_q) is block-diagonal, with blocks corresponding to the 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, Frobenius-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.)