Factorization of Shapovalov elements

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 $\beta$.Comment: 15 pages. Error corrected on p.1