TITLE: Bott's Formula for Grassmannians
QUESTION [5 upvotes]: Bott's Formula gives the dimension of the cohomology $H^{q}(\mathbb{P}^{n}, \Omega_{\mathbb{P}^{n}}^{p}(k))$ of the $k$-twisted sheaf of $p$-differential forms on the projective space $\mathbb{P}_{\mathbb{C}}^{n}$. I was wondering if there is a similiar formula for complex Grassmannians $\mathbb{G}(r, n)$ instead of projective spaces $\mathbb{P}_{\mathbb{C}}^{n}$. I searched for a while and I only found results that provide necessary and sufficient conditions for the vanishing of such cohomology groups. When such cohomology groups are non-zero I did not find any results providing the dimension of such cohomologies.
Do you know if there is any formula for the dimension of the cohomology groups $H^{q}(\mathbb{G}(r, n), \Omega_{\mathbb{G}}(k))$ as $\mathbb{C}$-vector spaces? Where could I find it?
Thank you very much!

REPLY [4 votes]: Yes, there are Bott's type formulas for Grassmannians.
For $r = 1$ in Lemma 0.1 of this paper
http://arxiv.org/pdf/alg-geom/9306010v2.pdf
In general you can look at:
D. Snow, Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces, Math. Ann. 276 (1986), 159-176.