If $M^{4k}$ is an oriented Riemannian $4k$-manifold, then the star-operator splits the bundle $\Lambda^{2k}$ into $\pm 1$-eigenspace bundles denoted $\Lambda^{2k}_{\pm}$.
I'm curious if anyone has ever computed $p_k(\Lambda^{2k}_{\pm})$ in terms of the Pontryagin classes and Euler class of the tangent bundle. (I'm still a bit sketchy on characteristic classes but from what I know this should be possible. Also, as a geometer I'm primarily thinking in terms of Chern-Weil theory.)
I know the answer when $k=1$ (when you integrate the class in question you get $2\chi(TM)\pm p_1(TM)$) but I played around with the $k=2$ case for a bit and ran into a hell of a computation, and I've done a bunch of searches and can't seem to find any relevant literature.
(I'd also be curious to know about Pontryagin classes of the other bundles of forms if anyone knows about them.)
