Proof of Kac and Rudakov's Conjecture on Generalized Verma Module over Lie Superalgebra E(5,10)

The exceptional infinite-dimensional linearly compact simple Lie superalgebra $E(5,10)$, which Kac believes, is the algebra of symmetries of the ${SU}_5$ Grand Unified Model. In this paper, we give a proof of Kac and Rudakov’s conjecture about the classification of all the degenerate generalized Verma module over $E(5,10)$. Also, we work out all the nontrivial singular vectors degree by degree. It is a potential that the representation theory of $E(5,10)$ will shed new light on various features of the the ${SU}_5$ Grand unified model.