Does anybody know if there is an analog of the Cartan (anti)involution for Walgebra associated to a nilpotent element e, which is principal in some Levi subalgebra of semisimple Lie algebra g? Actually, I am more interested whether there exists an analog of the Shapovalov form on a Verma module for such a Walgebra.
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In the affine case, there is a related discussion in a paper by KacWakimoto; see pp. 2325 in arXiv:mathph/0304011v2. 

