Characters of reductive groups over finite fields. (English) Zbl 0572.20026

Proc. Int. Congr. Math., Warszawa 1983, Vol. 2, 877-880 (1984).
[For the entire collection see Zbl 0553.00001.]
Let G be a connected reductive algebraic group defined over a finite field \(F_ q\) and \(G(F_ q)\) the finite group of \(F_ q\)-rational points of G. In his book [Characters of reductive groups over a finite field, Ann. Math. Stud. 107 (1984; Zbl 0556.20033)] the author gave a classification of the (complex) irreducible representations of \(G(F_ q)\) under the assumption that the center of G is connected. Here the author states that this assumption can be dropped and describes the classification. The description here is more in the spirit of Chapter 13 of [loc. cit.] rather than that of the Main Theorem 7.23; and the Langlands dual H of G over \({\mathbb{C}}\) is used, rather than a dual group \(G^*\) over \(F_ q\). The parameterization of the irreducible representations is in terms of certain finite sets \(\bar {\mathcal M}(g)\) associated with ”special” elements g in H.
Reviewer: B.Srinivasan


20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields