Number Theory Seminar
Apoorva Khare
Stanford
Faces and maximizer subsets of highest weight modules
Abstract: Verma modules over a complex semisimple Lie algebra, as well as their simple quotients are important and well-studied
objects in representation theory. We present three formulas to compute the set of weights of all such 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 $V^\mu$.
We also show that for all such simple modules, the convex hull of the weights is a $W_J$-invariant polyhedron for some parabolic
subgroup $W_J$. We compute its vertices, faces, and symmetries - more generally, we do so for all parabolic Verma modules, and
for all modules $V^\mu$ with $\mu$ not on a simple root hyperplane. Our techniques also enable us to completely classify
inclusions between "weak faces" of arbitrary $V^\mu$, in the process extending results of Vinberg, Chari, Cellini, and others
from finite-dimensional modules to all highest weight modules.
Please note special time, day, and seminar room.
Wednesday October 16, 2013 at 2:30 PM in SEO 1227