OK, I think I see the answer: by whatever character formula you like, you can see that for any fixed weight spaces (say, the $\lambda$ and $\lambda-n_\alpha\alpha$ for all $\alpha$), you can choose $\nu$ so that the multiplicities at all these points are very large compared to the differences between the those multiplicities. Since the number of elements in the crystal killed by $\tilde{F}\alpha^{n\alpha}$ \tilde{F}_{\alpha}^{n_\alpha}$ is just the difference in the weight multiplicities (or 0), the result follows by pigeonhole.
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OK, I think I see the answer: by whatever character formula you like, you can see that for any fixed weight spaces (say, the $\lambda$ and $\lambda-n_\alpha\alpha$ for all $\alpha$), you can choose $\nu$ so that the multiplicities at all these points are very large compared to the differences between the those multiplicities. Since the number of elements in the crystal killed by $\tilde{F}\alpha^{n\alpha}$ is just the difference in the weight multiplicities (or 0), the result follows by pigeonhole. |
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