On the Kernel of the affine Dirac operator

Let L be a finite-dimensional semisimple Lie algebra with a non-degenerate invariant bilinear form, \sigma an elliptic automorphism of L leaving the form invariant, and A a \sigma-invariant reductive subalgebra of L, such that the restriction of the form to A is non-degenerate. Consider the associated twisted affine Lie algebras L^, A^, and let F be the \sigma-twisted Clifford module over A^ associated to the orthocomplement of A in L. 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^-module and F, into irreducible A^-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.