The characters of finite reductive groups with a non-connected centre:: Applications to linear and unitary special groups -: Introduction

A first aim of this paper is to present an overview of results obtained by several authors on the characters of finite reductive groups with non-connected centre. We are particularly interested in problems directly linked to the non-connectedness of the centre. We emphasise on Gelfand-Graev and semisimple characters. A second aim is to study the influence of the non-connectedness of the centre on the theory of character sheaves. We study more precisely the family of character sheaves whose support meets the regular unipotent class: these are analogues of the semisimple characters. The last aim is the application of these results to finite reductive groups of type A, split or not (as for instance the special linear or special unitary groups). Whenever the cardinality of the finite field is large enough, we obtain a parametrization of the irreducible characters, a parametrization of the character sheaves, and we show that the characteristic functions of character sheaves are Fourier transforms of the irreducible characters (Lusztig's conjecture). This gives a theoretical algorithm for computing the character table of these groups.