On the realization of the Gelfand Character of a finite group as a twisted trace

Abstract

We show that the Gelfand character ${\chi }_G$ of a finite group $G$ (i.e. the sum of all irreducible complex characters of $G$ ) may be realized as a `` twisted trace'' $g\mapsto Tr({\rho }_g\circ T)$ for a suitable involutive linear automorphism $T$ of $L^2(G)$, where $(L^2(G),\rho )$ is the right regular representation of $G$. Moreover, we prove that under certain hypotheses $T(f)=f\circ L(f\in L^2(G)),$ where $L$ is an involutive antiautomorphism of $G.$ The natural representation $\tau$ of $G$ associated to the natural $L$-conjugacy action of $G$ in the fixed point set $Fi{x}_G(L)$ of $L$ turns out to be a Gelfand Model for $G$ in some cases. We show that $(L^2(Fi{x}_G(L)),\tau )$ fails to be a Gelfand Model if $G$ admits non trivial central involutions.

arXiv comment16 p. Preprint to be submitted to J. Group Theory