Since both members are holomorphic in the region {$Re(s) > -\delta$} and coincide for $Re(s) \gg 0$, they coincide in the region $Re(s) > -\delta$. Let $\beta \in \Omega^1(M)$. We can compute for $Re(s) \gg 0$, since $\iota_{\tilde{X}} \hat{\Pi} = 0$ by [DR17a],

$$
\begin{align*}
\mathrm{tr}_s^\beta \beta \iota_{\tilde{X}} (\mathcal{L}_{\tilde{X}} + s)^{-1} (\mathrm{Id} - \hat{\Pi}) e^{-\varepsilon \mathcal{L}_{\tilde{X}}} &= \mathrm{tr}_s^\beta \beta \iota_{\tilde{X}} G_{\chi, \varepsilon, s} \\
&= \int_0^\infty e^{-ts} \mathrm{tr}_s^\beta \beta \iota_{\tilde{X}} e^{-(t+\varepsilon)\tilde{X}} (\mathrm{Id} - \hat{\Pi}) \\
&= \int_0^\infty e^{-ts} \mathrm{tr}_s^\beta \beta \iota_{\tilde{X}} e^{-(t+\varepsilon)\tilde{X}},
\end{align*}
$$

where we could interchange the integral and the flat trace thanks to the bound obtained in Lemma C.1. Now the Atiyah-Bott trace formula [AB67] gives

$$
\mathrm{tr}_s^\beta \beta \iota_{\tilde{X}} e^{-(t+\varepsilon)\tilde{X}} = 0
$$

since $\tilde{X}$ vanishes at its critical points. By holomorphy this holds true for any $s$ such that $Re(s) > -\delta$. In particular if $\lambda > 0$ is small enough

$$
\mathrm{tr}_s^\beta \beta \iota_{\tilde{X}} \hat{Y} (\mathrm{Id} - \hat{\Pi}) e^{-\varepsilon \tilde{X}} = \frac{1}{2i\pi} \int_{|s|=\lambda} \mathrm{tr}_s^\beta \beta \iota_{\tilde{X}} \frac{(\mathcal{L}_{\tilde{X}} + s)^{-1}}{s} (\mathrm{Id} - \hat{\Pi}) e^{-\varepsilon \mathcal{L}_{\tilde{X}}} ds = 0,
$$

where $(\mathcal{L}_{\tilde{X}} + s)^{-1} = \hat{Y} + \frac{\hat{\Pi}}{s} + \mathcal{O}(s)$. Therefore Proposition 10.2 is proved in the case where $(E, \nabla)$ is the trivial bundle. The general case is handled similarly. $\square$

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UNIVERSITÉ PARIS-SUD, DÉPARTEMENT DE MATHÉMATIQUES, 91400 ORSAY, FRANCE

E-mail address: yann.chaubet@math.u-psud.fr

INSTITUT CAMILLE JORDAN (U.M.R. CNRS 5208), UNIVERSITÉ CLAUDE BERNARD LYON 1, BÂTIMENT BRACONNIER, 43, BOULEVARD DU 11 NOVEMBRE 1918, 69622 VILLEURBANNE CEDEX

E-mail address: dang@math.univ-lyon1.fr