Online References for Cartan Geometry

There is a series of four recorded lectures by Rod Gover introducing conformal geometry and tractor calculus. Tractor bundles are natural bundles equipped with canonical linear connections associated to $(\mathfrak{g}, H)$-modules. Tractor connections play the same role in general Cartan geometries that the Levi-Civita connection plays in Riemannian geometry; for general Cartan geometries the tangent bundle does not have a canonical linear connection.

There's also a set of introductory notes on conformal tractor calculus written by Rod Gover and Sean Curry.

If you have the book in your library, I would also suggest having a look at Cap & Slovak's Parabolic Geometries text. This is the modern bible on Cartan geometry, and parabolic geometries in particular. It is more terse than Sharpe, but also covers much more. Parabolic geometries are Cartan geometries modelled on $(\mathfrak{g}, P)$ where $\mathfrak{g}$ is semisimple and $P$ is a parabolic subgroup. Parabolic geometries include conformal, projective geometry, CR geometry, and many more geometries of interest.

In the parabolic setting representation-theoretic tools are often used to construct invariant differential operators. For instance, there are so-called BGG sequences of operators associated to irreducible representations, which in the flat case compute the same sheaf cohomology groups as the twisted de Rham sequence.

You can download some shorter text dealing with conformal geometries from Slovák's homepage One could download his book with Čap from the infamous Russian server (which appears to be down at the moment). I'm not sure whether the Sharpe's book is there as well.

I think a really good introductory text is the book Cartan for beginners by Ivey and Landsberg which doesn't really deal with Cartan geometries per se but rather teaches the Cartan method which, in a sense, is precisely the machinery that really makes the Cartan geometries work.

Tractor connections and tractor bundles are not really part of Cartan geometries but rather an independent (and in many cases equivalent) approach to study geometrical problems. In conformal geometry they were discovered by T. Y. Thomas in the mid twenties. See Thomas's structure bundle for conformal, projective and related structures. for details.