Finite fields, root systems, and orbit numbers of Chevalley groups (Q1266432)

Let \(G\) be a finite group of Lie type. \textit{G. Lusztig} [Characters of reductive groups over a finite field (1984; Zbl 0556.20033)] gave a parametrization of the characters in \(\text{Irr}_{\overline\mathbb{Q}_\ell}(G^*)\), where \(\overline\mathbb{Q}_\ell\) is the algebraic closure of the \(\ell\)-adic rationals and \(\ell\) is a prime different from the defining characteristic. In case the algebraic group \({\mathbf G}^*\) of \(G\) has connected centre (such as \(\text{SL}_n(q)\), \(\text{PGL}_n(q)\)), this parametrization is given in terms of irreducible unipotent characters of the centralizers \(G_s\) of the semisimple conjugacy classes \([S]\) of \(G\). The paper under review describes a combinatorial technique to determine the number of semisimple conjugacy classes and adjoint orbits with fixed class of centralizers for simply connected finite groups of Lie type.