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It may be better to think of $G/B^-$ as the variety of all Borel subgroups. Any $T$ fixed point $B$ on this variety will do. If one has such a fixed point and an equivariant line bundle, it becomes a puzzle to compare representations of $G$ and $B$, where $B$ acts on the fiber.
@AlexYoucis Yes, I was thinking of using 'power reductivity' (='adequate'), and the fact that Mumford's conjecture is now known in such generality that one does not need to consult Haboush or Seshadri when checking conditions.
You may find my expository paper Reductivity properties over an affine base, Indagationes Mathematicae (2020), doi.org/10.1016/j.indag.2020.09.009 helpful.
It is true that the assumption is that $k$ is a field. But this is not used here in an essential way. Anyway, the answer to what filtration you should use to reduce to the $\mathbb Z$ finite modules is: Use the Donkin truncation functors for the action of $SL_2$ from the left. So let the saturated set $\pi$ of dominant weights grow in Donkin 1986.
