Factorization of Shapovalov elements

TL;DR

This paper extends the explicit factorization of Shapovalov elements in universal enveloping algebras, providing a product formula for these elements for multiple roots and weights, except for a few special cases.

Contribution

It generalizes the known explicit matrix element approach for $m=1$ to cases where $m>1$, covering all roots except three exceptional cases in certain Lie algebras.

Findings

Explicit product formulas for $ heta_{eta,m}$ for most roots and weights.

Extension of matrix element approach from $m=1$ to $m>1$ cases.

Identification of exceptions in $rak{g}_2$, $rak{f}_4$, and $rak{e}_8$.

Abstract

Shapovalov elements $\theta_{\beta,m}$ are special elements in a Borel subalgebra of a classical or quantum universal enveloping algebra parameterized by a positive root $\beta$ and a positive integer $m$. They relate the canonical generator of a reducible Verma module with highest vectors of its Verma submodules. For $m=1$, they can be explicitly obtained as matrix elements of the inverse Shapovalov form. We extend this approach to $m>1$ for all $\beta$ but three roots in $\mathfrak{g}_2$, $\mathfrak{f}_4$, and $\mathfrak{e}_8$, presenting $\theta_{\beta,m}$ as a product of matrix elements of weight $\beta$.