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!