ON THE KERNEL OF THE AFFINE DIRAC OPERATOR

Let g be a finite-dimensional semisimple Lie algebra with a non-degenerate invariant bilinear form ((.),(.)), sigma an elliptic automorphism of g leaving the form ((.),(.)) invariant, and a a sigma-invariant subalgebra of g, such that the restriction of the form ((.),(.)) to a non-degenerate. Let (L) over cap (g, sigma) and (L) over cap (a, sigma) be the associated twisted affine Lie algebras and F(sigma)(p) the sigma-twisted Clifford module over (L) over cap (a, sigma), associated to the orthocomplement p of a in g. Under suitable hypotheses on sigma 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 (L) over cap (g, sigma)-module and F(sigma)(p), into irreducible (L) over cap (a, sigma)-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.