Classification of degenerate Verma modules for E(5,10)

Abstract

Given a Lie superalgebra $\frak g$ with a subalgebra $\frak g_{\geq 0}$, and a finite-dimensional irreducible $\frak g_{\geq 0}$-module $F$, the induced $\frak g$-module $M(F)=U({\frak g}) \otimes_{U(\frak g_{\geq 0})} F $ is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra ${\frak g}=E(5,10)$ with the subalgebra $\frak g_{\geq 0}$ of minimal codimension. This is done via classification of all singular vectors in the modules $M(F)$. Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as 'exceptional' de Rham complexes for $E(5,10)$.