On the asymptotics of Hecke operators for reductive groups

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