On the Schur indices of certain irreducible characters of reductive groups over finite fields (Q1112174)

Let G be a connected, reductive algebraic group defined over \(F_ q\), F a Frobenius endomorphism of G, and \(G^ F\) the finite group of F-fixed points of G. In a previous paper [Q. J. Math., Oxf. II. Ser. 32, 443-452 (1981; Zbl 0474.20022)] the author investigated the Schur indices of the irreducible characters of \(G^ F\) under the assumption that the center of G was connected. In this paper he drops that assumption. He studies the rationality properties of the characters \(\lambda^{G^ F}\) of \(G^ F\) induced from certain linear characters \(\lambda\) of a Sylow p- subgroup of \(G^ F\); the Gelfand-Graev characters of \(G^ F\) are special cases of these. Among the results proved is the following: Let q be odd. If \(\chi\) is of multiplicity one in some \(\lambda^{G^ F}\) then the rational Schur index \(m_ Q(\chi)\leq 2\). In particular it follows that if \(\chi\) is a Deligne-Lusztig character, i.e. of the form \(\pm R_ T^{\theta}\) where T is an F-stable maximal torus of G and \(\theta\) is a character of \(T^ F\), then \(m_ Q(\chi)\leq 2\). As an example the author calculates the local Schur indices of cuspidal Deligne-Lusztig characters of SL(n,q) where q is an even power of an odd prime.