Positivity on simple $G$-varieties

Let $X$ be a normal projective variety equipped with an action of a semisimple algebraic group $G$, and assume that $X$ contains a unique closed orbit. Let $B$ be a Borel subgroup of $G$ and let $E$ be a $B$-equivariant vector bundle on $X$. In this article, we prove that $E$ is ample (respectively, nef) if and only if its restriction to the finite set of $B$-stable curves in $X$ is ample (respectively, nef). Moreover, we compute the nef cone of the blow-up of a nonsingular simple $G$-projective variety $X$ at a unique $B$-fixed point $x^-$, referred to as the sink of $X$. As an application, when $X$ is nonsingular, we calculate the Seshadri constants of any ample line bundle (not necessarily $G$-equivariant) at $x^-$. In addition, we compute the Seshadri constants of $B$-equivariant vector bundles at $x^{-}$.