Spin Module Versions of Weyl's Reciprocity Theorem for Classical Kac-Moody Lie Algebras — An Application to Branching Rule Duality

We study tensor products of the spin modules (i.e. the Fermion Fock space representations) for classical (simple or affine) Kac-Moody Lie algebras. We find out that there are mutually commutant pairs of classical Kac-Moody algebras acting on the spin modules, and describe the irreducible decompositions in terms of Young diagrams. As applications, we obtain a simple explanation of Jimbo-Miwa's branching rule duality (i.e. isomorphisms between coset Virasoro modules) [JM], generalization thereof and the duality of the modular transformation rules of affine Lie algebra characters. Contents 0. Introduction 1. Preliminaries for Classical Lie Algebras 1.1. Conventions 1.2. Affine Kac-Moody Lie algebras 1.3. Modules which admit actions of outer automorphisms 1.4. Notations for classical simple Lie algebras 1.5. Notations for classical affine Lie algebras 2. Clifford Algebras and Spin Representations 2.1. Clifford algebra C(W) 2.2. The spin module /\(W~) of C(W) 2.3. The spin module /\ (Wj) of o(N) 2.4 The spin modules /\ (W N z+h ' ~) of o(N) A 3. Dual Pairs of Classical Lie Algebras 3.1. Definition of representations 3.2. The irreducible decompositions: The main result in the finite dimensional case (Theorem 3.2) 3.3. Proof of Theorem 3.2 4. Dual Pairs of Classical Affine Lie Algebras 4.1. Definition of representations 4.2. The irreducible decompositions: The main result in the affine Lie algebra case (Theorem 4.2) 4.3. Proof of Theorem 4.2: Case of h = 0 4.4. Proof of Theorem 4.2 continued: Case of h = 1/2 5. An Application to the Duality of Branching Rules 5.1. Seesaw pairs and dualities