Global section of universal bundle on Grassmanian

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.

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 $n=\dim V$).