Tensoring generalized characters with the Steinberg character

Abstract

Let G be a reductive connected algebraic group over an algebraic closure of a finite field of characteristic p. Let F be a Frobenius endomorphism on G and write G := G F for the corresponding finite group of Lie type. We consider projective characters of G in characteristic p of the form St ·β, where β is an irreducible Brauer character and St the Steinberg character of G. Let M be a rational G-module affording β on restriction to G. We say that M is G-regular if for every F -stable maximal torus T distinct weight spaces of M are nonisomorphic T -modules. We show that if M is G-regular of dimension d, then the lift of St · β decomposes as a sum of d regular characters of G.