Sur les caractères des groupes réductifs finis a centre non connexe: applications aux groupes spéciaux linéaires et unitaires
The main topic treated in this book is a study of the properties of finite reductive groups with disconnected centre. Let G be a connected reductive group over a finite field Fq of characteristic p with Frobenius map F. The Lustig conjecture relates almost characters of the finite reductive group GF to characteristic functions of character sheaves of G (up to a scalar). If the corresponding scalars can be computed, the conjecture makes it possible to compute irreducible characters of GF. The Lustig conjecture was proved by T. Shoji for the case when the group G has a connected centre and by J.-L. Waldspurger for Sp2n and On (with p and q big enough). In the book, a circle of related problems (a parametrization of irreducible characters, a parametrization of character sheaves and computations of the character table) are treated for the case when the centre of G is not connected. It contains a review of results obtained by several authors as well as new results. In particular, the (generalized) Lusztig conjecture is proved here for the special linear group and for the special unitary group in the case that p is arbitrary and q is large enough.