We study general highest weight modules V over a Kac-Moody algebra (one may assume this to be \sl(n) throughout the talk, without sacrificing novelty). We present a "first order" invariant for every V, which yields the convex hull of its weight-set, the Weyl group stabilizer of its character, and for certain V the weight-set itself.

 

This invariant has numerous applications; we will discuss two. First, it yields positive formulas (without cancellations) for the weights of all non-integrable simple highest weight modules over Kac-Moody algebras. For generic highest weights, we also present a weight-formula similar to the Weyl-Kac character formula. For the remaining highest weights, this formula fails in a striking way, suggesting the existence of "multiplicity-free" Macdonald identities for affine root systems.

 

Second, this first-order invariant helps study the convex hull of weights of all highest weight modules V, extending the notion of the Weyl polytope from integrable V to all V. We obtain a uniform description of the face lattice of this hull. This extends results by Satake [Ann. of Math. 1960], Borel-Tits [Publ. Math. IHES 1965], Cellini-Marietti [IMRN 2015], and others including Casselman and Vinberg - from finite-dimensional modules in finite type, to all modules in arbitrary type. (Partly joint with Gurbir Dhillon.)