Let $G=G(k, V)$ be the Grassmanian of $k$-dimensional subspaces of the $n$th dimensional vector space $V$, regarded as a smooth algebraic variety over $\mathbb{C}$. Denote with $S$ the tautological (universal) bundle over $G$.
How can I prove that $H^0(G, S^*) \simeq V^*$ and $H^0(G, V/S) \simeq V$?
EDIT: I found this result stated on a paper from Kapranov ("Coherent sheaves on Grasmann manifold"), who claims that "These facts are well known". Unfortunately, after lot of research, I could not find this statement in any reference where I looked for it. It is reasonable that the proof has to be done "by hands" like in the case of the tautological sheaf $\mathcal O (-1)$ over $\mathbb{P}^n$ but I can not solve this on my own.