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1) More care is needed about the meaning of tensor product of two irreducible representations in the Kac-Moody case, even when the modules involved are "integrable" and exhibit some of the classical character behavior. Tensoring infinite dimensional modules is quite a subtle question. (I'm not at all a specialist, however.) 2) As Emerton points out, you don't need Weyl's character formula in the classical case, but of course you do need well-behaved tensor products.
@Pete There was an older European tradition followed in the US of having an advisor look out for a position. But at UMass as elsewhere the program has changed and a tenure-track has for decades led to tenure (though earlier reviews have sometimes led people to look elsewhere). The tenure clock remains a problem for recruiting people who have already shown their value, here as elsewhere, since an "early" tenure case gets harsher scrutiny outside the department.
This is definitely a direction where exploration is needed, though I can't supply it. The older work on linear groups doesn't help (I only got the example in my ancient paper from Tits, who may have gotten it from Chevalley). In particular, my 1967 paper doesn't lead onward at all, though it is freely available online as other Pacific Journal papers are. Where I tried to use Lie algebras, the Frobenius kernels would be the obvious substitute to exploit for arbitrary powers of the prime.
As Steven implies, the language is too fuzzy at times. Some features of the tensor product of modules over $\mathbb{C}$ are easy to describe in terms of weights, but detailed module structure gets very complicated. The solution by Shrawan Kumar of the old PRV Conjecture (Parthasarathy, Range Rao, Varadarjan) in Invent. Math. 93 (1988) is a sample of this. "Extremal" weights in the irreducible case are just the Weyl group conjugates of the highest weight (all have multiplicity 1), but in a tensor product what is extreme/extremal? (And which Joseph paper do you refer to?)
This short paper is apparently not available online, but a version of Lazard's theorem is also written down in the 1970 book by Demazure-Gabriel, Groupes algebriques, I: see IV, section 4, 4.1.
References? Besides textbooks, it's interesting to look at some of the relevant literature such as MR943925 (89j:17009) 17B10 (22E46) Kumar, Shrawan (6-TIFR), Proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture. Invent. Math. 93 (1988), no. 1, 117–130.
Yes, this is the standard notion developed originally in the setting of finite dimensional irreducible representations of complex semisimple Lie algebras (or groups); extremal weights have multiplicity 1. This can to some extent be carried over to "integrable" representations of affine Kac-Moody algebras, where Kac found a good analogue of the Weyl character formula. Also, the theory of Demazure modules (geometrically motivated) involves study of the subspace obtained by fixing an extremal weight space and applying to it all negative root vectors in the Lie algebra.
