  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.
