Let $G$ be a sufficiently nice topological or Lie group (e.g. compact), and let $H$ be a closed subgroup. This data determines a principal $H$ bundle $G \rightarrow G/H$ defined by the projection $g \mapsto gH$.

In this case, we have Haar measures on $G$ and $G/H$ which allow us to perform invariant integration. I am wondering how this can be generalized to more general principal bundles.

Let $P$ be a principal $H$ bundle with automorphism group $\operatorname{Aut}(P)$. Is it possible to define an $\operatorname{Aut}(P)$-invariant measure on $P$, or otherwise define an invariant integral (e.g. using differential forms) of vector valued functions on P?

More specifically, I am interested in integrating over $P$ functions $f$ in the following space: $$ M = \{ f : P \rightarrow V \; | \; f(ph) = \rho(h^{-1}) f(p) \}. $$ Where $(\rho, V)$ is a representation of $H$. (Functions in $M$ are in one to one correspondence with sections of the associated vector bundle $P \times_\rho V$)

The automorphism group $\operatorname{Aut}(P)$ acts on $M$ by $f(p) \mapsto f(a^{-1} p)$ for $a \in \operatorname{Aut}(P)$. If I'm not mistaken, we have for the bundle $G \rightarrow G/H$ that the Haar measure is invariant to this action. Hence my question is whether there is such a thing as "generalized Haar measure" on more general principal bundles.

infinite dimensional. On which space is this group supposed to act? There are many possible choices. $\endgroup$ – Liviu Nicolaescu Dec 18 '18 at 20:51