Deligne-Lusztig restriction of Gelfand-Graev characters

Let G be a connected reductive algebraic group defined over a finite field Fq. 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(Fq) 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(Fq) using the theory of Kostant-Slodowy transversal slices for the nilpotent orbits of the Lie algebra of G.