  In this paper we study certain fundamental and distinguished subsets of
weights of an arbitrary highest weight module over a complex semisimple Lie
algebra. These sets ${\rm wt}_J \mathbb{V}^\lambda$ are defined for each
highest weight module $\mathbb{V}^\lambda$ and each subset $J$ of simple roots;
we term them "standard parabolic subsets of weights". It is shown that for any
highest weight module, the sets of simple roots whose corresponding standard
parabolic subsets of weights are equal form intervals in the poset of subsets
of the set of simple roots under containment. Moreover, we provide closed-form
expressions for the maximum and minimum elements of the aforementioned
intervals for all highest weight modules $\mathbb{V}^\lambda$ over semisimple
Lie algebras $\mathfrak{g}$. Surprisingly, these formulas only require the
Dynkin diagram of $\mathfrak{g}$ and the integrability data of
$\mathbb{V}^\lambda$. As a consequence, we extend classical work by Satake,
Borel-Tits, Vinberg, and Casselman, as well as recent variants by
Cellini-Marietti to all highest weight modules.
  We further compute the dimension, stabilizer, and vertex set of standard
parabolic faces of highest weight modules, and show that they are completely
determined by the aforementioned closed-form expressions. We also compute the
$f$-polynomial and a minimal half-space representation of the convex hull of
the set of weights. These results were recently shown for the adjoint
representation of a simple Lie algebra, but analogues remain unknown for any
other finite- or infinite-dimensional highest weight module. Our analysis is
uniform and type-free, across all semisimple Lie algebras and for arbitrary
highest weight modules.
