Singer torus in irreducible representations of GL (n, q)

Let G = PGL (n,q) be the projective general linear group of degree n over a finite field of q elements. Let T be a Singer torus of G, that is, a maximal torus of G of order vertical bar T vertical bar = (q(n) - 1)/(q - 1). Let chi be an irreducible character of G. We prove that the trivial character 1(T) of T occurs as a constituent of the restriction of chi to T unless chi(1) < vertical bar T vertical bar. This answers a question raised by Pablo Spiga.