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We prove that Zhu's algebra and the $C_2$-algebra of type ${\tt C}_m$ havethe same dimension, and we compute the graded character of the latter. Maximalparabolic subalgebras of the symplectic algebra play a central role in ourconstruction. For the orthogonal algebras our methods do not allow to describethe whole $C_2$-algebras, we get only a description of a certain quotient ofthe algebra.

Zhu's algebra and the $C_2$-algebra in the symplectic and the orthogonal cases