Title: Weights of highest weight modules over Kac–Moody algebras
Speaker: G V Krishna Teja (IISc Mathematics)
Date: 16 June 2022
Time: 1:30 pm
Venue: Microsoft Teams (online)
This thesis explores highest weight modules $V$ over complex semisimple and Kac-Moody algebras.
The first part of the talk addresses (non-integrable) simple highest weight modules $V = L(\lambda)$.
We provide a “minimum” description of the set of weights of $L(\lambda)$, as well as a “weak Minkowski
decomposition” of the set of weights of general $V$. Both of these follow from a “parabolic”
generalization of the partial sum property in root systems: every positive root is an ordered sum of
simple roots, such that each partial sum is also a root.
Second, we provide a positive, cancellation-free formula for the weights of arbitrary highest weight
modules $V$. This relies on the notion of “higher order holes” and “higher order Verma modules”, which
will be introduced and discussed in the talk.
Third, we provide BGG resolutions and Weyl-type character formulas for the higher order Verma modules
in certain cases - these involve a parabolic Weyl semigroup. Time permitting, we will discuss about weak
faces of the set of weights, and their complete classification for arbitrary $V$.