Modular realizations of hyperbolic Weyl groups

TL;DR

This paper explores the structure of hyperbolic Weyl groups through their isomorphisms with modular groups over integer domains in division algebras, providing new realizations and analysis methods.

Contribution

It introduces novel modular realizations of hyperbolic Weyl groups, especially W(E8), using integer domains in quaternions and octonions, and discusses automorphic form construction.

Findings

Established isomorphisms between hyperbolic Weyl groups and modular groups.

Provided a new realization of W(E8) using unit octavians.

Outlined methods for constructing automorphic forms for these groups.

Abstract

We study the recently discovered isomorphisms between hyperbolic Weyl groups and unfamiliar modular groups. These modular groups are defined over integer domains in normed division algebras, and we focus on the cases involving quaternions and octonions. We outline how to construct and analyse automorphic forms for these groups; their structure depends on the underlying arithmetic properties of the integer domains. We also give a new realization of the Weyl group W(E8) in terms of unit octavians and their automorphism group.