Global section of universal bundle on Grassmanian 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?
 A: These are simple instances of the Bott-Borel-Weil theorem. For a complex semsimple group $G$ and a parabolic subgroup $P$ and a complex irreducible representation $W$ of $P$ consider the homogeneous vector bundle $G\times_P W\to G/P$. In this situation the BBW theorem computes the cohomology of the shaef of local holomorphic sections of this bundle as a representation of $G$. The case you need here is that the highest weight of $W$ already is $G$-dominant and integral, in which case the cohomology is concentrated in degree zero and is the $G$-irreducible representation of the same highest weight. (Observe that $S^*$ is the $P$-irreducible quotient of $V^*$, while $V/S$ is the $P$-irreducible quotient of $V$.) 
The classical Borel-Weil theorem handles the case where $P=B$, the Borel subgroup of $G$, and states that the finite dimensional irredcible representations of $G$ corresponding to a dominant integral weight can be realized as the space of holomorphic sections of the homogeneous line bundle on the full flag manifold $G/B$ induced by the one-dimensional representation of $B$ defined by that weight. 
A nice exposition of the BBW-theorem can be found in the book on the Penrose transform by Baston and Eastwood. 
A: I don't know a reference either but one can argue as follows:
Let $U\subseteq V$ be of dimension $k$ and let $P$ be its stabilizer in $GL(V)$. Then the morphism $\pi:S=GL(V)\times^PU\to V$ is proper and surjective. Moreover one checks that all of its fibers are irreducible. The normality of $V$ implies $\pi_*\mathcal O_S=\mathcal O_V$, in particular, each global function on $S$ is a pull-back from $V$. Specializing to homogeneous functions of degree $1$ one gets $H^0(G,S^*)=V^*$. The other equality is obtained from the fact that $U\mapsto U^\perp=(V/U)^*$ yields an isomorphism between $Gr_k(V)$ and $Gr_{n-k}(V^*)$ (with $n=\dim V$).
