The Shapovalov determinant for the Poisson superalgebras

Among simple Z-graded Lie superalgebras of polynomial growth, there re seve ral which have no Cartan matrix but, nevertheless, have quadratic even Casi mir element C-2: these re the Lie superalgebra e(L)(1|6) of vector fields o n the (1|6) - dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra sh(0 \ 2k) of special Hamil tonian fields in 2k odd indeterminates, and the Kac-Moody version of sh(0|2 k). Using C-2 we compute N. Shapovalov determinant for e(L)(0|6) and sh(0|2 k), and for the Poisson superalgebras po(0|2k) associated with sh(0|2k). A. Shapovalov described irreducible finite dimensional representations of po( 0|n) and sh(0|n); we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over po(0|2k) and sh(0|2k).