Positivity of vector bundles on homogeneous varieties

We study the following question: Given a vector bundle on a projective variety $X$ such that the restriction of $E$ to every closed curve $C \,\subset\, X$ is ample, under what conditions $E$ is ample? We first consider the case of an abelian variety $X$. If $E$ is a line bundle on $X$, then we answer the question in the affirmative. When $E$ is of higher rank, we show that the answer is affirmative under some conditions on $E$. We then study the case of $X \,=\, G/P$, where $G$ is a reductive complex affine algebraic group, and $P$ is a parabolic subgroup of $G$. In this case, we show that the answer to our question is affirmative if $E$ is $T$--equivariant, where $T\, \subset\, P$ is a fixed maximal torus. Finally, we compute the Seshadri constant for such vector bundles defined on $G/P$.