What is the Induced Representation in Geometric Terms

As is well known, for $G$ a Lie group, and $H$ a subgroup of $G$ such that $G/H$ is homogeneous space (or maybe this is always a homogeneous space?), we have a correspondence between representations of $H$ and equivariant vector bundles over $G/H$. Moreover, for ${\cal E}_{\pi}$ the vector bundle corresponding to a rep $\pi$, we have a rep of $G$ on the smooth sections $\Gamma^{\infty}({\cal E})$ of $\cal E$. Is this what peopole call the rep of $G$ induced by $\pi$?

-
Why the algebraic-geometry tag? – Martin Brandenburg Mar 12 '13 at 16:30
+1 I don't know how to answer your actual question, but doesn't $G$ act smoothly and transitively on $G/H$ so that it is homogenous? – Alex Youcis Mar 12 '13 at 16:31
The short answer is IMVHO "Yes". You probably want to be sure that $G/H$ has a rich enough structure for the stuff about bundles to make sense. @Martin: I think that algebraic-geometry is an appropriate tag. If $G, H$ and $G/H$ can be given a structure of algebraic varieties, then the machinery of algebraic geometry (sheaf cohomology for example) comes to the fore. This is the case often enough, and there are parallels in the Lie group side (some might say that the stuff in the AG side consists of analogues of tools from the Lie group side!). – Jyrki Lahtonen Mar 12 '13 at 16:55