IIA and IIB spinors from K(E10)

We analyze the decomposition of recently constructed unfaithful spinor representations of K(E10) under its SO(9)×SO(9), and SO(9)×SO(2) subgroups, respectively, where K(E10) is the 'maximal compact' subgroup of the hyperbolic Kac-Moody group E10. We show that under these decompositions, respectively, one and the sameK(E10) spinor gives rise to both the fermionic fields of IIA supergravity, and to the (chiral) fermionic fields of IIB supergravity. This result is thus the fermionic analogue of the decomposition of E10 under its SO(9,9) and SL(9)×SL(2) subgroups, respectively, which yield the correct bosonic multiplets of (massive) IIA and IIB supergravity. The essentially unique Lagrangian for the supersymmetric E10/K(E10)σ-model therefore can also capture the dynamics of IIA and IIB including bosons and fermions in the known truncations.