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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.

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

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