Suppose $G$ is a Lie group and $H$ is a (closed) subgroup. There is a natural $H$ principal bundle $H \to G \to G/H$. This master bundle seems to lead to the following vector bundle $$ V \to E \to G/H. $$ I have heard it said (cf. Nicholas Kuhn's answer) that vector bundles of this sort are completely classified by irreps $\mu$ of the "little group" $H$ and the bundle is given by $$ E = G \times_\mu V \simeq \{ \psi: G \to V: \psi(x g) = \mu(g^{-1})\psi(x) \}. $$ The right hand side is just the induced representation $\text{Ind}_H^G(\mu)$.
Question 1: Is this true? If so, can someone provide an elementary proof (or cite one)?
Suppose the previous is affirmative. I have also heard it said that the previous space is equipped with a natural flat connection (descending from a flat connection on $G \times G/H$), see Dan Ramras' answer.
Question 2: Is this basically the statement of the Riemann-Hilbert correspondence (flat connections on $E$ are in correspondence with the fundamental group $\pi_1(G/H)$)?
On the other hand, suppose $G = SU(2)$ and $H = U(1)$ so that $G/H \simeq S^2$. Then $$ \mathbb{C} \to SU(2) \times_n \mathbb{C} \to S^2 $$ is a twisted vector bundle over the $2$-sphere such that $n \in \mathbb{Z}$ indexes the irreps of $U(1)$. However, there is a non-flat connection on this space (that seems fairly natural) given by equation C.19 in Particle on the sphere: group-theoretic quantization in the presence of a magnetic monopole .
Question 3: This isn't really a math question per say but is there some "natural" connection on this bundle (i.e., is the flat one or the non-flat one more "natural", maybe induced from some intermediate principal $U(n)$ bundle)?
