Weyl Denominator Identity for Affine Lie Superalgebras with Non-zero Dual Coxeter Number

Weyl denominator identity for the affinization of a basic Lie superalgebra with non-zero Killing form was formulated by V. Kac and M. Wakimoto and was proven by them for the defect one case. In this paper we prove this identity. 0. Introduction Let g be a basic Lie superalgebra with a non-zero Killing form. Let ĝ be the affinization of g. Let h (resp., ĥ) be the Cartan subalgebra in g (resp., in ĝ) and let (−,−) be the bilinear form on ĥ which is induced by the Killing form on g. Let ∆ (resp., ∆̂) be the root system of g (resp., of ĝ). We set ∆ := {α ∈ ∆0| (α, α) > 0}. Then ∆ is a root system of a simple Lie algebra. Let ∆̂ be the affinization of ∆. Denote by Ŵ (resp., W) the subgroup of GL(ĥ) generated by the reflections sα : α ∈ ∆̂0, (α, α) > 0 (resp., sα : α ∈ ∆ ). Then W is the Weyl group of ∆ and Ŵ is the corresponding affine Weyl group. Recall that Ŵ = W ⋉ T , where T ⊂ Ŵ is the translation group, see [K2], Chapter 6. Let Π be a set of simple roots for g, and let Π̂ = Π ∪ {α0} be the corresponding set of simple roots for ĝ. Let ∆+, ∆̂+ be the corresponding sets of positive roots. We set R := ∏ α∈∆+,0 (1− e) ∏ α∈∆+,1 (1 + e−α) , R̂ := ∏ α∈∆̂+,0 (1− e) ∏ α∈∆̂+,1 (1 + e−α) . Following [KW], we call R the Weyl denominator and R̂ the affine Weyl denominator. The Weyl denominator identity conjectured by V. Kac and M. Wakimoto in [KW] can be written as R̂e = ∑ w∈T w(Re), where ρ̂ ∈ ĥ is such that 2(ρ̂, α) = (α, α) for each α ∈ Π. The original form of this identity is given in formula (2). In this paper we prove this identity. Supported in part by ISF Grant No. 1142/07. 1