Specialized characters of affine superalgebras
Let \({\mathfrak g}(A)\) be the super analogue of the Kac-Moody algebra with symmetrizable Cartan matrix \(A\). In [\textit{V. G. Kac}, Adv. Math. 30, 85-136 (1978; Zbl 0391.17010)] a super version of the Weyl-Kac character formula for irreducible highest weight modules enables to compute the principal specialization of such characters. This specialization corresponds to the principal gradation of \({\mathfrak g}(A)\) in which simple root vectors are of first degree. In the paper of Kac this formula was given only for the non-super case. The purpose of the paper under review is to establish the appropriate super version when the principal gradation of \({\mathfrak g}(A)\) assigns degree 1 to the odd simple roots and degree 2 to the even ones. Then the author computes the specialized characters of the fundamental highest weight modules of the affine superalgebras of rank 1, namely \(A^{(2)}(0,1)\cong C^{(2)}(2)\), \(B^{(1)}(0,1)\) and \(A^{(4)}(0,2)\). Finally, the author gives the characters of the representations of the superalgebras \(A^{(4)}(0,2l)\) and \(A^{(2)}(0,2l-1)\) described in his earlier paper [\textit{G. Golitzin}, J. Algebra 117, No. 1, 198-226 (1988; Zbl 0652.17013)].
scientific article; zbMATH DE number 1271652