Abstract

We show that the Gelfand character χ G \chi-{G} of a finite group (i.e.The sum of all irreducible complex characters of ) may be realized as a "twisted trace"g Tr (ρ g T) g\mapsto\operatorname{Tr}(\rho-{g}\circ T) for a suitable involutive linear automorphism of L 2 (G) L^{2}(G), where (L 2 (G), ρ) (L^{2}(G),\rho) is the right regular representation of . Moreover, we prove that, under certain hypotheses, we have T (f) = f L T(f)=f\circ L (f L 2 (G) f\in L^{2}(G)), where is an involutive anti-Automorphism of . The natural representation of associated to the natural-conjugacy action of in the fixed point set Fix G (L) \operatorname{Fix}-{G}(L) of turns out to be a Gelfand model for in some cases. We show that (L 2 (Fix G (L)), τ) (L^{2}(\operatorname{Fix}-{G}(L)),\tau) fails to be a Gelfand model if admits non-Trivial central involutions.