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I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...

For a homogeneous space $G/H$ a homogeneous vector bundle has a total space of the form $G \times_{\rho} V$, where $(V,\rho)$ is a representation of $H$ and $G \times_{\rho} V$ is the set of ...

Let $G$ be a Lie group equipped with a left-invariant Riemannian metric, and let $M$ be a submanifold of $G$ containing the identity $e\in G$. It is not difficult to show that, if the tangent bundle $...