  The representation theory of involutory (or 'maximal compact') subalgebras of
infinite-dimensional Kac-Moody algebras is largely terra incognita, especially
with regard to fermionic (double-valued) representations. Nevertheless, certain
distinguished such representations feature prominently in proposals of possible
symmetries underlying M theory, both at the classical and the quantum level.
Here we summarise recent efforts to study spinorial representations
systematically, most notably for the case of the hyperbolic Kac-Moody algebra
$E_{10}$ where spinors of the involutory subalgebra $K(E_{10})$ are expected to
play a role in describing algebraically the fermionic sector of $D=11$
supergravity and M theory. Although these results remain very incomplete, they
also point towards the beginning of a possible explanation of the fermion
structure observed in the Standard Model of Particle Physics.
