On the Kernel of the Affine Dirac Operator

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