Construction of Kac–Moody superalgebras as minimal graded Lie superalgebras and weight multiplicities for Kac–Moody superalgebras

We construct a Kac–Moody superalgebra ℒ as the minimal graded Lie superalgebra with local part V*⊕ge⊕V, where ???? is a “smaller” Lie superalgebra inside ℒ, V is an irreducible highest weight ????-module, and V* is the contragredient of V. We show that the weight multiplicities of irreducible highest weight modules over Kac–Moody superalgebras of finite type and affine type [more precisely, Kac–Moody superalgebras of type B(0,r), B(1)(0,r), A(4)(2r,0), A(2)(2r−1,0), and C(2)(r+1)] are given by polynomials in the rank r. The degree of these weight multiplicity polynomials are less than or equal to the depth of weights.