  We give the first positive formulas for the weights of every simple highest
weight module $L(\lambda)$ over an arbitrary Kac-Moody algebra. Under a mild
condition on the highest weight, we also express the weights of $L(\lambda)$ as
an alternating sum similar to the Weyl-Kac character formula.
  To obtain these results, we show the following data attached to a highest
weight module are equivalent: (i) its integrability, (ii) the convex hull of
its weights, (iii) the Weyl group symmetry of its character, and (iv) when a
localization theorem is available, its behavior on certain codimension one
Schubert cells. We further determine precisely when the above datum determines
the weights themselves. Moreover, we use condition (iv) to relate localizations
of the convex hull of the weights with the introduction of poles of the
corresponding $D$-module on certain divisors, which answers a question of
Brion.
  Many of these results are new even in finite type. We prove similar
assertions for highest weight modules over a symmetrizable quantum group.
