Higher spin representations of maximal compact subalgebras of simply laced Kac–Moody algebras

Given the maximal compact subalgebra k(A) of a split-real Kac–Moody algebra g(A) of type A, we study certain finite-dimensional representations of k(A) that do not lift to the maximal compact subgroup K(A) of the minimal Kac–Moody group G(A) associated to g(A) but only to its spin cover Spin(A) described in [11]. Currently, four elementary of these so-called spin representations are known. We study their (ir)reducibility, semisimplicity, and lift to the group level. The interaction of these representations with the spin-extended Weyl group is used to derive a partial parametrization result of the representation matrices by the real roots of g(A).