Suppose you have a scheme $X$ that is acted on by a torus $T$. Then the action induces a grading on the functions on $X$ by the character lattice of $T$. So for a fixed character $\lambda$, we can consider $\mathcal{O}_{X,\lambda}$, the $\lambda$ graded part. Assuming the quotient $X/T$ exists, these graded parts should descend to quasicoherent sheaves on the quotient.

My question is, when are these sheaves line bundles?

In the basic examples I know, they are always line bundles. For example, if you take $X= \mathbb{A}^{n+1} - 0$ and $T = \mathbb{C}^*$, then on $X$ you get the ordinary grading by homogeneous degree. When you descend to the quotient, you get the line bundles $\mathcal{O}(k)$ on $\mathbb{P}^n$. You can also take $G$ a complex semi-simple group, $B$ a Borel subgroup, $U$ the maximal unipotent. Then $G/U \rightarrow G/B$ is a torus quotient, and the graded pieces descend to line bundles. I think, a similar story is true for all homogeneous spaces, but I'm having a little trouble phrasing it in terms of torus quotients.

In fact, in these situations, these are all the line bundles.

So more generally, my question is, what properties can you require of the general $X$ so it behaves like the two examples above?