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Let $M$ be a Riemannian manifold. Assume that a finite group $G$ acts on $M$ as isometry. How can one prove that $G$ takes harmonic forms to harmonic forms?

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Let $(M,g)$ be an orientable Riemannian manifold. The Hodge Laplacian is defined by $\Delta = d d^* + d^* d$.

The differential $d$ commutes with pullback by an arbitrary smooth function. The codifferential, being dependent on the metric through the Hodge star, commutes with pullback by an isometry. Thus, $\Delta$ commutes with pullback by isometries.

To show that $d^*$ commutes with pullback by isometries, show that the Hodge star $*$ commutes with pullback by orientation preserving isometries. This in turn can be checked in the linear setting, on the exterior algebra of an inner product vector space with an orientation for a linear orientation preserving isometry. The codifferential $d^*$ uses the Hodge star twice, so in fact, it commutes with arbitrary isometries, orientation preserving or reversing.