Covers of tori over local and global fields

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 $T_{\Bbb{A}}$ when $T_{\Bbb{A}}$ 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.