\v{S}apovalov elements for simple Lie algebras and basic classical simple Lie superalgebras

Abstract

Let $M(\gl)$ be a Verma module for a basic classical simple Lie superalgebra $\fg \neq G(3)$ defined using the distinguished Borel subalgebra, and let $\gc$ be an isotropic positive root of $\fg.$ As a special case of our first main result we show that if $\mu, \gl \in \fh^*$ with $\gl-\mu = \gc$ we have $$\dim \Hom_{\sfg}(M(\mu),M(\gl))\le 1.$$ This result applies to the construction of \v{S}apovalov elements for isotropic roots. The proof rests on a comparison with the corresponding result for a certain simple Lie algebra $G$.