**Date & Time:** Monday, December 15, 2014, 14:30-15:30.

**Venue:** Ramanujan Hall

**Title:** : Faces and supports of highest weight modules

**Speaker:** Apoorva Khare, Stanford University

**Abstract: ** We present three formulas to compute the set of weights of all simple highest weight modules (and others) over a complex semisimple Lie algebra $\mathfrak{g}$. These formulas are direct and do not involve cancellations. Our results extend the notion of the Weyl polytope to general highest weight $\mathfrak{g}$-modules $\mathbb{V}^\mu$.

We further classify and describe the vertices, faces, and their symmetries for a very large class of highest weight modules, including all parabolic Verma modules and their simple quotients. Finally, we completely classify inclusions between faces of arbitrary $\mathbb{V}^\mu$, in the process extending results of Vinberg, Chari, Cellini, and others from finite-dimensional modules to all highest weight modules.