Let $M$ be a smooth, closed manifold, equipped with a smooth (finite) triangulation $K$. Further, let $H$ be a Hilbert space, $G := \pi_1(M)$ and let $\rho: G \to GL(H)$ be a representation (with $GL(H)$ denoting the group of (continuous) automorphisms of $H$). Via the action of $\rho$, we can therefore regard $H$ as a (left-) $\mathbb C[G]$-module with.

The triangulation $K$ lifts to a $G$-equivariant smooth triangulation $\widetilde{K}$ on the universal cover $\widetilde{M}$ of $M$. The associated simplical cochain complex $C(\widetilde{K})$ (with complex coefficients) can thus be regarded as a finite complex of (right-) $\mathbb C[G]$-modules, so we can therefore form the tensor product complex $C(\widetilde{K},H) := C(\widetilde{K}) \otimes_{\mathbb C[G]} H$, a cochain complex of Hilbert spaces with continuous boundary maps. Denote by $\Delta_\rho$ the associated (combinatorial) Laplacian.

We can also consider the flat canonical bundle $H_{\rho}$ over $M$ associated to $\rho$ with $\Gamma(H_\rho)$ its associated space of smooth sections and $$\Omega(M,H_\rho) := \Gamma(H_\rho) \otimes_{C^\infty(M,\mathbb C)} \Omega(M)$$ the cochain complex of $H_\rho$-valued differential forms over $M$.

A choice of Riemannian metric $g$ on $M$ and Hermitian form $h$ on $H_{\rho}$ give rise, in the usual way, to an inner product structure on $\Omega (M,H_{\rho})$, formal adjoints to the exterior derivatives and, thus, to a Laplacian $\Delta(g,h): \Omega(M,H_{\rho}) \to \Omega(M,H_{\rho})$. Denote by $\Omega_{(2)}(M,H_\rho)$ the $L^2$-completion of the pre-Hilbert space $\Omega(M,H_{\rho})$

Question: Does there exist a reasonable map of Hilbert-space complexes $A: \Omega_{(2)}(M,H_\rho) \to C(\tilde{K},H)$ inducing an isomorphism of sub-complexes $\tilde{A}: \ker(\Delta(g,h)) \to \ker(\Delta_{\rho})$?

When $\rho$ is the well-known, extensively studied regular representation $\rho: G \to \mathcal N(G)$ (so that $H = l^2(G)$), the above is a classic result from Dodziuk (See: J. Dodziuk. de Rham-Hodge theory for L2-cohomology of infinite coverings. Topology, 16(2):157–165, 1977).

Somehow, it seems to be, more or less, common knowledge that his result extends to arbitrary representations on Hilbert spaces, but neither have I seen an explicit proof of this general case, nor it is clear to me how his specific methods extend to the general case. I'd be perfectly happy with a reference.


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