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In this paper we study certain fundamental and distinguished subsets ofweights of an arbitrary highest weight module over a complex semisimple Liealgebra. These sets ${\rm wt}_J \mathbb{V}^\lambda$ are defined for eachhighest 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 anyhighest weight module, the sets of simple roots whose corresponding standardparabolic subsets of weights are equal form intervals in the poset of subsetsof the set of simple roots under containment. Moreover, we provide closed-formexpressions for the maximum and minimum elements of the aforementionedintervals for all highest weight modules $\mathbb{V}^\lambda$ over semisimpleLie algebras $\mathfrak{g}$. Surprisingly, these formulas only require theDynkin 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 byCellini-Marietti to all highest weight modules.  We further compute the dimension, stabilizer, and vertex set of standardparabolic faces of highest weight modules, and show that they are completelydetermined by the aforementioned closed-form expressions. We also compute the$f$-polynomial and a minimal half-space representation of the convex hull ofthe set of weights. These results were recently shown for the adjointrepresentation of a simple Lie algebra, but analogues remain unknown for anyother finite- or infinite-dimensional highest weight module. Our analysis isuniform and type-free, across all semisimple Lie algebras and for arbitraryhighest weight modules.

Standard parabolic subsets of highest weight modules