Foncteurs de Lusztig et caractères des groupes linéaires et unitaires sur un corps fini. (Lusztig functors and characters of linear and unitary groups over a finite field)
Let \({\mathcal G}\) be a reductive algebraic group defined over \(F_ q\). Let \({\mathcal L}\) be a Levi subgroup of a parabolic subgroup \({\mathcal P}\), such that \({\mathcal L}\) (but not necessarily \({\mathcal P})\) is defined over \(F_ q\). Let G, L denote the finite groups of rational points of \({\mathcal G}\), \({\mathcal L}\) respectively. Lusztig has defined a map \(R^ G_ L\) (more precisely, \(R^ G_{L,P}\); this map is here called the Lusztig functor) from the Grothendieck group R(L) of L into R(G). The first part of this paper is devoted to a study of the map \(R^ G_ L\). In particular, a character formula (3.2) and a ''theorem of Curtis type'' (3.5) are proved. Now let \({\mathcal G}=GL_ n\), and \(G_ n=GL(n,q)\) or U(n,q). \textit{A. V. Zelevinsky} [Representations of Finite Classical Groups (Lect. Notes Math. 869, 1981; Zbl 0465.20009)] introduced a Hopf algebra structure on \(RG=\oplus_{n\geq 0}R(G_ n)\) when \(\hat G_ n=GL(n,q)\) and studied the representations of \(G_ n\) via the algebra RG. The authors study RG in the more general case in the second part of the paper. A structure theorem for the Hopf algebra RG is proved in 9.5. Finally they study a ''Hall submodule'' of R(G)\(\otimes Q\) which is of interest in computing the characters of \(G_ n\).
scientific article; zbMATH DE number 4008665
scientific article