  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 Lusztig one by a linear character, and we
relate this linear character to some stabilizers. Some consequences on
characteristic functions of character sheaves are obtained. In the forthcoming
part II, we will apply these methods to obtain a precision on Digne, Lehrer and
Michel's theorem on Lusztig restriction of Gel'fand-Graev characters.

Group Theory;Representation Theory