On realizations of the Gelfand character of a finite group

Abstract

We show that the Gel’fand character χG of a finite group G (i.e. the sum of all irreducible complex characters of G ) may be realized as a “ twisted trace” g 7→ T r(ρg ◦ T ) for a suitable involutive linear automorphism of L 2 (G), where ρ stands for the right regular representation of G in L 2 (G). We prove further that, under certain hypotheses, T may be obtained as T (f) = f ◦L, where L is an involutive antiautomorphism of the group G so that T r(ρg ◦ T ) = |{h ∈ G : hg = L(h)}|. We also give in the case of the group G = P GL(2, Fq) a positive answer to a question of K. W. Johnson asking whether it is possible to express the Gel’fand character χG as a polynomial in a single irreducible character η of G.