On the kernel of the affine Dirac operator

Let g be a finite-dimensional semisimple Lie algebra and (· , ·) its Killing form, σ an elliptic automorphism of g, and a a σ-invariant reductive subalgebra of g, such that the restriction of the form (· , ·) to a is non-degenerate. Let L̂(g, σ) and L̂(a, σ) be the associated twisted affine Lie algebras and F σ(p) the σ-twisted Clifford module over L̂(a, σ), associated to the orthocomplement p of a in g. Under suitable hypotheses on σ and a, we provide a general formula for the decomposition of the kernel of the affine Dirac operator, acting on the tensor product of an integrable highest weight L̂(g, σ)-module and F σ(p), into irreducible L̂(a, σ)-submodules. As an application, we derive the decomposition of all level 1 integrable irreducible highest weight modules over orthogonal affine Lie algebras with respect to the affinization of the isotropy subalgebra of an arbitrary symmetric space.