Deligne–Lusztig characters for finite reductive monoids

Abstract

The theory of reductive monoids has been well developed by Renner and the author, cf. [8, 13]. This paper concerns the finite reductive monoids M^F where M is a reductive monoid and F[ratio ]M→M is a Frobenius map. We show that the Deligne–Lusztig characters R_{T, θ} of finite reductive groups G^F have natural extensions to M^F. We accomplish this by first showing that the action of a finite monoid (by partial transformations) on an algebraic variety gives rise to a virtual character of the monoid via étale cohomology. We then find the correct analogue of the algebraic set L⁻¹(U) (where L is the Lang map) for M. The action of M^F on this algebraic variety gives rise to virtual characters R_{T¯, θ} where T is an F-stable maximal torus of G, T¯ the Zariski closure of T in M and θ an irreducible character of the finite commutative monoid T¯^F. We go on to show that any irreducible character of M^F is a component of some R_{T¯, θ}.