Corollary 3

The map which associates to any continuous central function $\varphi$ on G the function $w(G)^{-1/2}J(\rho )(\varphi |T)$ induces an isomorphism from the space of central representative functions on G to the space of anti-invariant elements of $\Theta (T)$; it extends by continuity to an isomorphism of Hilbert spaces from ZL$^2(G)$ to the subspace of anti-invariant elements of $L^2(T)$.