Affiliation: ISI Bangalore
Title: Weights of highest weight modules over Borcherds-Kac-Moody algebras
Date and Time: March 11, 2024, 11:00:00 Hours
Venue: A-369
Asbtract: $\mathfrak g$ is a general complex simple/ Kac-Moody/ Borcherds-Kac-Moody (BKM) Lie algebra (not always symmetrizable), $\mathfrak{h}$ its Cartan subalgebra, and $V$ (any) highest weight $\mathfrak{g}$-module with highest weight $\lambda$. The weight-sets wt$V$ of $V$ are seemingly unknown even for integrable $V$ over BKM $\mathfrak{g}$; in KM case not beyond: 1) integrable (simple) and parabolic Vermas $V$ classically; (thereby) 2) non-integrable simple $V$ due to Dhillon and Khare.
This talk presents our recent explicit, non-recursive, uniform (over $\mathfrak{g},\lambda, V$) formulas for wt$V$ for all modules $V$ (integrable to any non-simple, and $\forall\ \lambda$) -- extending formulas for $V$ in cases 1), 2) above -- over all KM (Joint work with Khare) and BKM $\mathfrak{g}$ (with Pal). Interestingly, our weight-formulas appear as Weyl (sub)group-orbits, similar to integrable~$V$.
Key ingredients in our formulas: 1)~A novel concept of holes in modules $V$:- all monomials acting locally nilpotently on $V$ (generalizing "Chevalley-Serre" relations for simple $V$) 2)~Suitable notions of dominant integral $\lambda$ and (new) integrable modules $V$ in BKM case. 3) Studying $V$ over $\mathfrak{g}$ generated by 3-dim. Heisenbergs. We do not know characters of $V$ in cases 2) not covered by Weyl-Kac-Borcherds character formulas) and 3). Curious by-products of our formulas: {\it higher order Vermas} over KM $\mathfrak{g}$, yielding wt$V$ $\forall V$.