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