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 $G$.

On Kapranov's "Coherent sheaves on Grasmann manifold" the following result are stated:

$H^0(G, S^*) \simeq V^*$ and $H^0(G, V/S) \simeq V$

The author claims that "These facts are well known".

However after a 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 $\mathbb{P}^n$.

Do you have any suggestion?