Take a vector space $V$ (finite dimensional, over the complex numbers), let $G=SL(V)$. The group $G$ acts on $\mathbb{P}V$ and we can linearize its action to an action on the line bundle $\mathcal{O}(1)$. This gives an action on $H^0(\mathbb{P}V,\mathcal{O}(1))$. My question is


as $G$-module, is $H^0(\mathbb{P}V,\mathcal{O}(1))$ isomorphic to $V$ or to $V^{\vee}$?? and why??



Example of linearization for GIT
By definition $\mathbb{P} V = \mathrm{Proj}\ \mathrm{Sym}(V^\vee)$. Quasi-coherent sheaves on the $\mathrm{Proj}$ are identified with graded modules over the graded ring (modulo torsion). Under this correspondence the sheaf $\mathcal{O}(1)$ goes to the graded module $\mathrm{Sym}(V^\vee)[1]$ (the ring shifted down by 1). The functor of global sections corresponds to taking the degree 0 part of the module, which is just $V^\vee$.


