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functor to move $\lambda$ around inside a Weyl chamber, or else pass to the closure (using the fact that each weight lies in the upper closure of a unique chamber); then apply the functor to those Verma modules …

This kind of result suggests strongly that for finite dimensional $G_r$-modules the best possible upper bound should be something like $r$ times the dimension of the nilpotent variety. … Roughly speaking, complexity 0 corresponds to projective modules whereas the largest complexity for a simple module tends to occur for the trivial module. …

In particular, James exploited the fact that the characteristic 0 Specht modules have a fairly natural reduction mod $p$ for any prime $p$. … This is somewhat analogous to the algebraic group situation, where "Weyl modules" come by such reductions and then have a unique distinguished composition factor. …

[ADDED] If you want concrete examples showing that $N(\lambda)$ need not be the sum of Verma modules, you have to introduce more notation and do some careful bookkeeping with weights and Weyl group elements … For all integral weights in suitable Weyl chambers (and some singular weights on walls), one then gets examples of Verma modules which you ask about. …

modules. … Obviously what is going on in such examples requires specific primes for the specific groups involved as well as specific simple or projective modules. …