The highest weight representations of the contact Lie superalgebra on 1|6-dimensional supercircle

Among simple Z-graded Lie superalgebras of polynomial growth there is only one without Cartan matrix but with an invariant nondegenerate supersymmetric bilinear form. This is kL(1|6), the Lie superalgebra of vector fields on the (1|6)-dimensional supercircle preserving the Pfaff equation α = 0, where α = dt + Σ1≤i≤3 (ξi,dηi + ηidξ) and where ξ, η are the odd variables, t exp(iφ) for the angle parameter φ on the circle. For kL(1|6) we compute the Casimir element wherefrom we deduce the Shapovalov determinant and the description of the irreducible Verma modules over kL(1|6). (author)