Stabilizer groups of Yang-Mills connections Let $G$ be a compact Lie group with complexification $G^c$, and consider a principal $G^c$-bundle $P^c \to M$ together with a reduction $P \subseteq P^c$ to $G$. Assume that $M$ is a Riemann surface.
Then every connection on $P$ gives a unique holomorphic structure on $P^c$, and accordingly we get a natural action of the gauge group $Gau(P^c) = \Gamma(P^c \times_{G^c} G^c)$ on the space $C(P)$ of connections of $P$. One may think of $Gau(P^c)$ as the complexification of the gauge group $Gau(P)$ of $P$.
Question: Is there a relation between the stabilizer $Gau(P^c)_A$ and the stabilizer $Gau(P)_A$ of a Yang-Mills connection $A$ on $P$? At least on the level of their Lie algebras?

Background: According to Atiyah-Bott, the Yang-Mills functional can be viewed as the momentum map squared of the $Gau(P)$-action on $C(P)$ with respect to a natural symplectic structure. On the other hand, in Hessians of the Calabi Functional and the Norm Function it was shown that the Hessian of the momentum map squared yields a pair of commuting operators on the complexified Lie algebra. These operators then give a relation between the stabilizers of the complexified action and the original action. Applying this to the above Yang-Mills setting yields the following conjecture:
For a Yang-Mills connection $A$, the Lie algebra $gau(P^c)_A$ of $Gau(P^c)_A$ decomposes as
$$
gau(P^c)_A = \bigoplus_{\lambda > 0} E_\lambda \, \, \oplus (gau(P)_A)^{c},
$$
where $E_\lambda$ are $\lambda$-eigenspaces of $2 i [\star F_A, \cdot]$ and $(gau(P)_A)^{c}$ is the complexification of the Lie algebra of the stabilizer $Gau(P)_A$.
 A: For a Yang-Mills connection $A$, one indeed has a decomposition
$$H_A\bigl(Ad P \otimes \mathbb{C}\bigr) = \bigl(gau(P)_A\bigr)_{\mathbb{C}} \oplus \bigoplus_{\lambda > 0} H_A\bigl(Ad_\lambda P\bigr),$$
where $H_A$ denotes the space of holomorphic sections (with respect to the holomorphic structure induced by the connection $A$) and $Ad_\lambda P$ are the eigenspaces of the endomorphism $2 i \, [\star F_A, \cdot]$ on $Ad P$.
In fact, this follows from observations made by Atiyah and Bott in their paper "The Yang-Mills equations over Riemann surfaces". On page 556 they show that the endomorphism $2 i \, [\star F_A, \cdot]$ on $Ad P$ has constant eigenvalues and thus $Ad P \otimes \mathbb{C}$ decomposes into eigenbundles $Ad_\lambda P$ for $\lambda \in \mathbb{R}$. This decomposition then induces decompositions on the level of differential forms:
\begin{equation}
    \Omega^k(M, Ad P \otimes \mathbb C) = \bigoplus_{\lambda} \Omega^k(M, A_\lambda P).
\end{equation}
As a consequence of the Yang-Mills equation, the operators $\bar{\partial}_A$ and $2 i \, [\star F_A, \cdot]$ commute.
Hence,
\begin{equation}
    H_A\bigl(Ad P \otimes \mathbb C\bigr) = \bigoplus_{\lambda} H_A\bigl(Ad_\lambda P\bigr) \, .
\end{equation}
By considering appropriate Laplacian operators, one can show that $H_A\bigl(Ad_\lambda P\bigr)$ is isomorphic to $\bigl(gau(P)_A\bigr)_{\mathbb C}$ for $\lambda = 0$ and is trivial for $\lambda < 0$, see Lemma 5.9 (iii) and p. 559 in Atiyah and Bott.
