Deligne–Lusztig restriction of Gelfand–Graev characters

Let G be a connected reductive algebraic group defined over a finite field . In [F. Digne, G.I. Lehrer, J. Michel, On Gelʹfand–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 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 using the theory of Kostant–Slodowy transversal slices for the nilpotent orbits of the Lie algebra of G.