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Let $G$ be a compact connected Lie group and $P$ a principal $G$-bundle over a finite CW complex $X$. The gauge group $\mathcal{G}(P)$ is defined to be the group of principal bundle automorphisms of $P$ covering the identity map on $X$. Alternatively, $\mathcal{G}(P)$ is the space of sections of the adjoint bundle $(P\times G)/G$, where $G$ acts the second factor by conjugation.

Question: How can we compute the number of connected components of $\mathcal{G}(P)$? I am particularly interested in the special case where $X=H/K$, $H$ and $K$ both compact and connected Lie groups of equal ranks, $G=U(n)$, and $P=H\times_K U(n)$ with $K$ acting on $U(n)$ by left multiplication through a unitary representation of $K$ of dimension $n$: \begin{eqnarray}k\cdot g=\rho(k)g\ \text{for }k\in K, g\in U(n), \rho\text{ an }n\text{-dimensional unitary representation.}\end{eqnarray} I would like to see if $\mathcal{G}(P)$ is connected if $\rho$ has trivial subrepresentation of sufficiently large multiplicity. In fact, it is known that $\mathcal{G}(P)$ is connected for $\rho$ being trivial and $n$ sufficiently large because $[H/K, U(\infty)]=K^{-1}(H/K)=0$.

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    $\begingroup$ The pullback to $H$ of this principal bundle is a trivial $U(n)$-bundle over $H$. The group of sections of this trivial $U(n)$-bundle is isomorphic to the group of continuous functions $Hom(H,U(n))$. There is an induced action of $K$ on $Hom(H,U(n))$. The group of sections of the original principal bundle are the same as the group of $K$-invariants of the induced action of $K$ on $Hom(H,U(n))$. $\endgroup$ – Jason Starr May 8 '17 at 12:33

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