Construction of Kac-Moody superalgebras as minimal graded Lie superalgebras and weight multiplicities for Kac-Moody superalgebras (Q2738093)

Let \(\mathfrak g\) be a Kac-Moody Lie superalgebra given by a (generalized) Cartan matrix and \(V\) an irreducible highest weight \(\mathfrak g\)-module. The authors construct a new Kac-Moody superalgebra \(L\) as the minimal graded Lie superalgebra with the local part \(V^*\oplus{\mathfrak g}^e\oplus V\), where \(V^*\) is the contragredient of \(V\), and \({\mathfrak g}^e=\mathfrak g\) if \(\mathfrak g\) is of affine type and \({\mathfrak g}^e={\mathfrak g}\oplus \mathbb C K\) otherwise, where the element \(K\) acts trivially on \({\mathfrak g}^e\). NEWLINENEWLINENEWLINEAs the main result the authors show that the weight multiplicities of irreducible highest weight modules over some Kac-Moody superalgebras of finite type and affine type are given by polynomials in the rank \(r\). Namely, this result is established for Kac-Moody superalgebras of types \(B(0,r)\), \(B^{(1)}(0,r)\), \(A^{(4)}(2r,0)\), \(A^{(2)}(2r-1,0)\), and \(C^{(2)}(r+1)\). The degree of these weight multiplicity polynomials are less than or equal to the depth of weights.