Apoorva Khare : Faces and maximizer subsets of highest weight modules

Posted by Ramin Takloo-Bighash , part of the Number Theory Seminar.

At

Oct. 16, 2013, 2:30 p.m.

In

SEO 1227

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.

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