Sunada unitary equivalence without a finite heat trace ($L^2(N{)}^{U_i}={⨁}_{\rho }{V}_{\rho }^{U_i}\otimes {\mathcal{H}}_{\rho }$)

A finite isometry group decomposes $L^2(N)$ into its irreducible representation factors $V_{\rho }$ and multiplicity Hilbert spaces ${\mathcal{H}}_{\rho }$. The Laplacian commutes with the group, so it acts as the identity on each representation factor. The averaging projection gives $\dim V_{\rho }^{U_i}=|U_i{|}^{-1}{\sum }_{u\in U_i}{\chi }_{\rho }(u)$. For Gassmann equivalent subgroups these dimensions agree. Choosing unitary identifications of the invariant finite-dimensional factors therefore intertwines the quotient Laplacians, even when a noncompact covering manifold has no finite heat trace.