TITLE: Cartan involution for finite W-algebras
QUESTION [12 upvotes]: Does anybody know if there is an analog of the Cartan (anti)involution for W-algebra
associated to a nilpotent element e, which is principal in some Levi subalgebra
of semi-simple Lie algebra g? Actually, I am more interested whether there exists an analog of the Shapovalov form on a Verma module for such a W-algebra.

REPLY [2 votes]: In the affine case, there is a related discussion in a paper by Kac-Wakimoto; 
see pp. 23-25 in arXiv:math-ph/0304011v2.