Factorization of Shapovalov elements

Shapovalov elements $θ_{β,m}$ are special elements in a Borel subalgebra of a classical or quantum universal enveloping algebra parameterized by a positive root $β$ 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 $β$ but three roots in $\mathfrak{g}_2$, $\mathfrak{f}_4$, and $\mathfrak{e}_8$, presenting $θ_{β,m}$ as a product of matrix elements of weight $β$.

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