The Weyl law for algebraic tori

TL;DR

This paper provides an asymptotic count of automorphic characters of algebraic tori based on their analytic conductor, linking local Langlands correspondence with counting automorphic representations.

Contribution

It introduces a new asymptotic evaluation method for automorphic characters of algebraic tori using the analytic conductor defined via the local Langlands correspondence.

Findings

Asymptotic formula for automorphic characters count

Connection between analytic conductor and automorphic representation counting

Framework applicable to reductive groups

Abstract

We give an asymptotic evaluation for the number of automorphic characters of an algebraic torus $T$ with bounded analytic conductor. The analytic conductor which we use is defined via the local Langlands correspondence for tori by choosing a finite dimensional complex algebraic representation of the $L$-group of $T$. Our results therefore fit into a general framework of counting automorphic representations on reductive groups by analytic conductor.