Let $G$ be a (real) Lie group.
Notation: For $X$ a smooth manifold:
- we denote by $\mathrm{Bun}_G(X)$ the groupoid of smooth principal $G$-bundles on $X$;
- we denote by $\mathrm{Core}(\mathrm{Vect}(X))$ the core groupoid (the wide subcategory consisting of all isomorphisms) of the category $\mathrm{Vect}(X)$ of finite-rank real smooth vector bundles on $X$.
By a "representation" of $G$ we always mean a finite-dimensional real linear Lie group representation $(V, \rho : G\rightarrow \mathrm{GL}_{\mathbb{R}}(V))$. Given such a representation of $G$, we denote by $\phi_{\rho;X} : \mathrm{Bun}_G(X) \rightarrow \mathrm{Core}(\mathrm{Vect}(X))$ the corresponding "$\rho$-associated bundle construction morphism". Note $\phi_{\rho;X}$ is a functor / morphism between groupoids.
Definition: we say a given representation $\rho$ of $G$ is "special" if for every smooth manifold $X$, there exist a (faithful) sub-groupoid $\mathcal{A}_X \subset \mathrm{Core}(\mathrm{Vect}(X))$ and a functor/groupoid-morphism $\xi_{\rho;X} : \mathrm{Bun}_G(X) \rightarrow \mathcal{A}_X$, such that:
$\xi_{\rho;X}: \mathrm{Bun}_G(X) \rightarrow \mathcal{A}_X$ is an equivalence of groupoids; and
$\phi_{\rho;X} \cong \iota \circ \xi_{\rho;X}$ where $\iota : \mathcal{A}_X \hookrightarrow \mathrm{Core}(\mathrm{Vect}(X))$ is the inclusion/forgetful functor; and
$\mathcal{A}_X$ is functorial, and $\xi_{\rho;X}$ is natural, in the smooth manifold $X$.
My questions are: Does there exist a Lie group $G$ which admits no "special" representation (as in the above Definition)? What about a compact Lie group?
For example, if $G = \mathrm{O}(n)$ and $\rho$ is the defining representation on $\mathbb{R}^n$, then if I'm not mistaken, we can take
- $\mathcal{A}_X$ as the category of rank-$n$ (smooth) real vector bundles equipped with (smooth) bundle metric, with morphisms being vector bundle isomorphisms which preserve the bundle metric; and
- $\xi_{\rho;X}$ equips a $\rho$-associated vector bundle $E \rightarrow X$ with the unique per-fiber metric / inner product determined by the $O(n)$-action on the fibers of $E$. (Such an inner product exists, e.g. by applying the usual method of using Haar integration, and compactness of $G$, to obtain a $G$-invariant inner product.)