Mean Values and Quantum Variance for Degenerate Eisenstein Series of Higher Rank

Abstract

We investigate the mean value of the inner product of squared $\mathrm{GL}_{n}$ degenerate maximal parabolic Eisenstein series against a smooth compactly supported function lying in a restricted space of incomplete Eisenstein series induced from a $\mathrm{SL}_{2}(\mathbb{Z})$ Hecke-Maass cusp form $\varphi$. Our result breaks the fundamental threshold with a polynomial power-saving beyond the pointwise implications of the generalised Lindel\"{o}f hypothesis for $L$-functions attached to $\varphi$. Furthermore, we evaluate the archimedean quantum variance and establish approximate orthogonality, expanding upon Zhang's (2019) work on quantum unique ergodicity for $\mathrm{GL}_{n}$ degenerate maximal parabolic Eisenstein series as well as Huang's (2021) work on quantum variance for $\mathrm{GL}_{2}$ Eisenstein series. Despite the theoretical strength of these manifestations, our argument relies exclusively on the Watson-Ichino-type formula for incomplete Eisenstein series of type $(2, 1, \ldots, 1)$ and Jutila's (1996) asymptotic formula for the second moment of $L$-functions attached to $\varphi$ in long intervals, supplemented by a standard analytical toolbox.