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Suppose $(M,g)$ is a compact Riemannian manifold and $(E, \nabla, \lambda, B)$ is the following data:

1) $E$ is a complex vector bundle over $M.$

2) $\nabla$ is a flat connection.

3) $B$ is a symmetric, non-degenerate, parallel bi-linear form.

4) $\lambda$ is a "Cartan involution," that is, $\lambda$ is an anti-linear involution on $E$ such that $H(u,v):=-B(u,\lambda(v))$ is a Hermitian inner product. Here $u,v$ are sections of $E.$

Then given $\alpha \otimes u$ and $\beta\otimes v$ differential $k$-forms with values in $E,$ (so $u,v$ are sections of $E),$ then form the $L^2$-pairing via

\begin{align} (\alpha\otimes u, \beta\otimes v):=\int_{M} \alpha\wedge *\beta \ H(u,v) \end{align}

Here $*$ is the Hodge star operator given by the metric $g$ on $M.$

With respect to the metric $H,$ the flat connection splits as $\nabla=\nabla^{H}+\Psi$ where $\nabla^{H}$ is an $H$-unitary connection and $\Psi$ is a one-form with values in Hermitian endomorphisms of $E.$ Then we can form the exterior covariant derivative associated to $\nabla$ given by $d^{\nabla}=d^{H}+\Psi$ where here $\Psi$ acts on $E$-valued $k$-forms by wedging the forms and acting on the section via the endomorphism part of $\Psi.$ Using the $L^2$-pairing we can construct the Hodge Laplacian \begin{align} d^{\nabla}\delta^{\nabla}+\delta^{\nabla}d^{\nabla}, \end{align} where throughout $\delta$ denotes the formal adjoint of $d.$ Expanding this out using $\delta^{\nabla}=\delta^{H}+\Psi^{*}$ we obtain, \begin{align} d^{\nabla}\delta^{\nabla}+\delta^{\nabla}d^{\nabla}=d^{H}\delta^{H}+\delta^{H}d^{H} +\Psi\Psi^{*}+\Psi^{*}\Psi+d^H \Psi^{*}+ \Psi^{*}d^{H}+ \delta^{H}\Psi+ \Psi\delta^{H}. \end{align} In the above line juxtaposition stands for composition of operators.

I've found it claimed in a few places in a similar context that

\begin{align} d^H \Psi^{*}+ \Psi^{*}d^{H}+ \delta^{H}\Psi+ \Psi\delta^{H}=0. \end{align}

Nonetheless, after calculating for hours on end, I can't seem to work this out. It's highly likely that I'm doing something wrong, but now I'm getting nervous that perhaps I've misread the context where I previously found this formula stated.

Thus, if there's anyone that can verify that this formula is true in this context I would be very grateful. I've seen a couple papers claim that this is contained in the Annals paper of Matsushima and Murakami entitled "On vector bundle valued harmonic forms and automorphic forms on symmetric Riemannian manifolds." Nonetheless, I haven't been able to find this calculation in that paper, and even still I believe the context is a bit different. Thank you very much for your help.

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