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

Abstract

Among simple $\mathbb{Z}$-graded Lie superalgebras of polynomial growth there is only one without Cartan matrix but with an invariant nondegenerate supersymmetric bilinear form. This is $\mathfrak{k}^{\text{L}} (1|6)$, the Lie superalgebra of vector fields on the (1|6)-dimensional supercircle preserving the Pfaff equation α = 0, where $\alpha = dt + \sum\nolimits_{1 \leqslant i \leqslant 3} {(\xi _i {\text{d}}\eta _i + \eta _i {\text{d}}\xi _i )}$ and where ξ, η are the odd variables, t = exp(iφ) for the angle parameter φ on the circle. For $\mathfrak{k}^{\text{L}} (1|6)$ we compute the Casimir element wherefrom we deduce the Shapovalov determinant and the description of the irreducible Verma modules over $\mathfrak{k}^{\text{L}} (1|6)$.