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* ⊕ g^e ⊕ V, where g is a "smaller" Lie superalgebra inside ℒ, V is an irreducible highest weight g-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.