Is there an analogue of Kac-Weisfeiler conjecture (a theorem of Premet) on the power of p dividing dimension of an irreducible representation for Lie algebras of Cartan type? Benkart and Feldvoss mention this as an open problem (Problem 10) in their 2015 survey, but they don't seem to state a precise conjecture.
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$\begingroup$ Simple Lie algebras of Cartan type in prime characteristic are quite hard to study in a uniform way, unlike the Lie algebras of simple algebraic groups (even when these Lie algebras aren't quite simple). As far as I know, the representations have been studied only in some very special cases, for example by Feldvoss and Nakano. But some of the same questions do arise, even without the groups in the background. Sasha Premet should comment further on this, but meanwhile you might consider adding some tags such as 'lie-algebras'. $\endgroup$– Jim HumphreysNov 30, 2017 at 1:20
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$\begingroup$ P.S. Here is a link to the online version of the Feldvoss-Nakano paper I mentioned: sciencedirect.com/science/article/pii/S0021869397973439 $\endgroup$– Jim HumphreysNov 30, 2017 at 1:27
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$\begingroup$ Another useful link: mathscinet.ams.org/mathscinet-getitem?mr=3453057 (which is probably the same as the arXiv second version front.math.ucdavis.edu/1503.06762); but note in Problem 10(a) that they don't really mean "lower bound". $\endgroup$– Jim HumphreysNov 30, 2017 at 17:24
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