Deligne–Lusztig restriction of Gelfand–Graev characters

Abstract

Let G be a connected reductive algebraic group defined over a finite field F q . In [F. Digne, G.I. Lehrer, J. Michel, On Gel'fand–Graev characters of reductive groups with disconnected centre, J. Reine Angew. Math. 491 (1997) 131–147], it is proved that the Deligne–Lusztig restriction of a Gelfand–Graev character of the finite group G ( F q ) is still a Gelfand–Graev character. However, an ambiguity remains on the Gelfand–Graev character obtained. In this paper, we describe the Deligne–Lusztig restrictions of the Gelfand–Graev characters of the finite group G ( F q ) using the theory of Kostant–Slodowy transversal slices for the nilpotent orbits of the Lie algebra of G.