specific copy of $G$ in the configuration space. Above in (\[leftact\]) we had the distinguished action $\lambda:G\to\aut C(G)$ by $$\lambda_g(f)(h) := f(g^{-1} h) \, \, , \, \,
f \in C(G),\; g,h\in G.$$ The generalized Schrödinger representation $(T,U)$ above in (\[GSRp\]) is a covariant representation for the action $\lambda:G\to\aut C(G)$ so it is natural to take for our field algebra the crossed product C\*-algebra $C(G)\rtimes_\lambda G $ whose representations are exactly the covariant representations of the $C^*$-dynamical system defined by $\lambda$. The algebra $C(G)\rtimes_\lambda G $ is also called the generalised Weyl algebra, and it is well–known that $C(G)\rtimes_\lambda G\cong\cl K.\big(L^2(G)\big)$ cf. [@Rief] and Theorem II.10.4.3 in [@Bla1]. In fact $\pi_0\big(C(G)\rtimes_\lambda G\big)=\cl K.\big(L^2(G)\big)$ where $\pi_0:C(G)\rtimes_\lambda G\to\cl B.( L^2(G))$ is the generalized Schrödinger representation. Since the algebra of compacts $\cl K.\big(L^2(G)\big)$ has only one irreducible representation up to unitary equivalence, it follows that the generalized Schrödinger representation is the unique irreducible covariant representation of $\lambda$ (up to equivalence). Moreover, as $\psi_0$ is cyclic for $\cl K.\big(L^2(G)\big)$, the generalized Schrödinger representation is unitary equivalent to the GNS–representation of the vector state $\omega_0$ given by $\omega_0(A):={(\psi_0,\pi_0(A)\psi_0)}$ for $A\in
C(G)\rtimes_\lambda G$.

Note that the operators $U_g$ and $T_f$ in equation (\[GSRp\]) are not compact, 