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 an $\mathrm{SL}_{2}(\mathbb {Z})$ Hecke–Maaß cusp form $\varphi$. Our result breaks the fundamental threshold with a polynomial power-saving beyond the pointwise implications of the generalized Lindelöf hypothesis for L-functions attached to $\varphi$. Furthermore, we evaluate the Archimedean quantum variance and establish approximate orthogonality, expanding upon Zhang’s work on quantum unique ergodicity for $\mathrm{GL}_{n}$ degenerate maximal parabolic Eisenstein series as well as Huang’s 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 asymptotic formula for the second moment of L-functions attached to $\varphi$ in long intervals, supplemented by a standard analytical toolbox.