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 $[H/K, U(\infty)]=K^{-1}(H/K)=0$.