Contemporary Mathematics Volume 153, 1993 EXTENDING STEINBERG CHARACTERS WALTER FElT Dedicated to Bob Steinberg on the occasion of his 70th birthday May 25, 1992 ABSTRACT. let G be a semi simple group of Lie type over a finite field of char- acteristic p. Let () be the Steinberg character of G. If G J H it is known that ()extends to a character x of H. The purpose of this paper is to evaluate x on p' -elements. The problem is considered in a more general context and various properties of Steinberg characters are discussed. §1 Introduction. Let G(q) be a simply connected semi-simple group of Lie type over F q, where q is a power of the prime p. Let Sta(q) = St be the Steinberg character of G(q). See (St1, 15.5]. Suppose that G(q) 1 H. Then it is known that St extends to a character of H. See (Sc1], (Sc2], (M]. The main object ofthis paper is to get some information about the values of this extended character. This information is contained in Theorem C. We first need some definitions and preliminary results. Let p be a prime and let G be a finite group. A Steinberg character with respect to p or a p-Steinberg character of G is an irreducible character 8 of G such that 8(x) = ±lCa(x)lv for every p'-element x in G. Here Ca(x) denotes the centralizer of x in G and np is the p-part of the integer n. Since 8(1) = IGip for a p-Steinberg character e, it has p-defect 0 and so 8(y) = 0 for every p-singular element yin G. Suppose that p is a prime and G 1 H are finite groups. A relative ( G, H) p-Steinberg character is an irreducible rational valued character 8 of H such that 8(x) = ±ICa(x)lp for every p'-element x in H. Observe that if p f IH : Gl, then a relative (G, H) p-Steinberg character is a p-Steinberg character of H. However, in the general case the values of a 1991 Mathematics Subject Classification. Primary 20C15, Secondary 20C25, 20G40. This paper is in final form and no version of it will be submitted for publication elsewhere. 1 © 1993 American Mathematical Society 0271-4132/93 $1.00 + $.25 per page http://dx.doi.org/10.1090/conm/153/01303
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