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Abstract: Let G 0 be a connected unipotent algebraic group over a finite field F q, andlet G be the unipotent group over an algebraic closure F of F q obtained fromG 0 by extension of scalars. If M is a Frobenius-invariant character sheaf onG, we show that M comes from an irreducible perverse sheaf M 0 on G 0, which ispure of weight 0. As M ranges over all Frobenius-invariant character sheaves onG, the functions defined by the corresponding perverse sheaves M 0 form a basisof the space of conjugation-invariant functions on the finite group G 0F q,which is orthonormal with respect to the standard unnormalized Hermitian innerproduct. The matrix relating this basis to the basis formed by irreduciblecharacters of G 0F q is block-diagonal, with blocks corresponding to theL-packets of characters, or, equivalently, of character sheaves.We also formulate and prove a suitable generalization of this result to thecase where G 0 is a possibly disconnected unipotent group over F q. Ingeneral, Frobenius-invariant character sheaves on G are related to theirreducible characters of the groups of F q-points of all pure inner forms ofG 0.



Author: Mitya Boyarchenko

Source: https://arxiv.org/







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