# On the asymptotics of Hecke operators for reductive groups

## TL;DR

This paper investigates the asymptotic behavior of traces of Hecke operators for automorphic representations on split reductive groups over , providing explicit estimates and conditions under which classical and exceptional groups follow expected asymptotics.

## Contribution

It establishes the asymptotic formulas for Hecke operator traces on general split reductive groups, extending known results to broader classes under specific intertwining operator conditions.

## Key findings

- Asymptotic formulas derived for Hecke operator traces
- Explicit remainder estimates provided
- Conditions verified for classical and G_2 groups

## Abstract

In this paper, we study the asymptotic behavior of the traces of Hecke operators for spherical discrete automorphic representations of fixed level on general split reductive groups over $\mathbb{Q}$. Under a condition on the analytic behavior of intertwining operators, which is known for the classical groups and the exceptional group $G_2$, we obtain the expected asymptotics in terms of the spherical Plancherel measure and an explicit estimate for the remainder.