Résumé
In the first part, we construct a new isomorphism between the endomorphism
algebra of an induced cuspidal character sheaf and the group algebra of the
relative Weyl group involved. We show it differs from the isomorphism of
Lusztig by a linear character, and we relate this linear character to some
stabilizers. Some consequences for characteristic functions of character
sheaves are obtained. In the second part, we compute explicitly this linear
character whenever the cuspidal local system is supported by the regular
unipotent class and, as an application of these methods, we obtain a refinement
of Digne, Lehrer and Michel's theorem on Lusztig restriction of Gel'fand-Graev
characters.