A vertical Sato-Tate law for \(\mathrm{GL}(4)\) (Q6564136)

In recent work, Goldfeld, Stade, and Woodbury proved a Kuznetsov trace formula for \(GL(4)\) and used it to obtain an orthogonality relation for spherical Maass cusp forms. In the paper under review, the author extends this orthogonality relation to general Maass cusp forms and as an application obtains a doubly-weighted vertical Sato--Tate law for \(GL(4)\) with explicit error terms.\N\NThe first main result is Theorem 1.1, which is an extension to general Fourier coefficients of Theorem 1.1.1 [\textit{D. Goldfeld} et al., Forum Math. Sigma 9, Paper No. e47, 83 p. (2021; Zbl 1478.11078)]. In exchange for a slightly weaker error term, the author drops their assumption of the Ramanujan conjecture and instead uses a Ramanujan-on-average bound based on [\textit{V. Chandee} and \textit{X. Li}, Adv. Math. 365, Article ID 107060, 39 p. (2020; Zbl 1458.11076)].\N\NThe second main result is the weighted vertical Sato-Tate law given in Theorem 1.2, which follows from Theorem 1.1 and Theorems 8.4 and 9.1 in [\textit{F. Zhou}, Ramanujan J. 35, No. 3, 405--425 (2014; Zbl 1326.11021)]. The proof of the rate of convergence essentially follows Zhou's argument for the \(n = 3\) case and is an application of Theorem 1.1 and the Casselman-Shalika formula.