Modular realizations of hyperbolic Weyl groups

Abstract

We study the recently discovered isomorphisms between hyperbolic Weyl groups and modular groups over integer domains in normed division algebras. We show how to realize the group action via fractional linear transformations on generalized upper half-planes over the division algebras, focusing on the cases involving quaternions and octonions. For these we construct automorphic forms, whose explicit expressions depend crucially on the underlying arithmetic properties of the integer domains. Another main new result is the explicit octavian realization of $W+(E_{10})$, which contains as a special case a new realization of $W+(E_8)$ in terms of unit octavians and their automorphism group.