Covers of tori over local and global fields

TL;DR

This paper extends Langlands' work by describing automorphic and local representations of metaplectic tori, including spherical Hecke algebras and multiplicity-one results, within the Brylinski-Deligne framework.

Contribution

It provides a comprehensive description of irreducible and automorphic representations of covers of tori, including new results on Hecke algebras and multiplicity estimates.

Findings

Description of spherical Hecke algebras in the unramified local setting

Global multiplicity estimate for automorphic representations of covers of split tori

Proof of a multiplicity-one theorem for automorphic representations of covers of split tori

Abstract

Langlands has described the irreducible admissible representations of $T$, when $T$ is the group of points of an algebraic torus over a local field. Also, Langlands described the automorphic representations of $T_{\mathbb A}$ when $T_{\mathbb A}$ is the group of adelic points of an algebraic torus over a global field $F$. We describe irreducible (in the local setting) and automorphic (in the global setting) $\epsilon$-genuine representations for covers of tori, also known as metaplectic tori, which arise from a framework of Brylinski and Deligne. In particular, our results include a description of spherical Hecke algebras in the local unramified setting, and a global multiplicity estimate for automorphic representations of covers of split tori. For automorphic representations of covers of split tori, we prove a multiplicity-one theorem.