Higher Spin Representations in Kac-Moody-Theory

Abstract

For maximal compact subalgebras k(A) of split-real Kac-Moody algebras g(A), where A is a generalized Cartan matrix of arbitrary but simply-laced type, there exist four different types of finite-dimensional representations which are known as the 1/2-spin, 3/2-spin,...,7/2-spin representation respectively. In this work, the higher spin representations (3/2 and higher) are studied and described in terms of the 1/2-spin representation and representations of the Weyl group W(A). It is verified that they are spinorial in nature, meaning that they lift only to the spin group Spin(A) but not to the maximal compact subgroup K(A) of the minimal split-real Kac-Moody group G(A). The irreducible factors of the 3/2-spin representation and the 5/2-spin representation are determined for the example k(E10) both analytically as well as with computer-based methods. For arbitrary but simply-laced A a general criterion concerning their decomposition into irreducible factors is derived in terms of A and W(A). It is shown that certain tensor products of the irreducible factors mentioned above with the 1/2-spin representation are again irreducible. Finally, for A of untwisted affine type an inverse system of finite-dimensional representations of k(A) is constructed whose projective limit provides a faithful representation of k(A).