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

We investigate the mean value of the inner product of squared GLₙ degenerate maximal parabolic Eisenstein series against a smooth compactly supported function lying in a restricted space of incomplete Eisenstein series induced from an SL₂(Z) Hecke–Maaß cusp form φ. 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 φ. Furthermore, we evaluate the Archimedean quantum variance and establish approximate orthogonality, expanding upon Zhang’s work on quantum unique ergodicity for GLₙ degenerate maximal parabolic Eisenstein series as well as Huang’s work on quantum variance for GL₂ 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, … , 1) and Jutila’s asymptotic formula for the second moment of L-functions attached to φ in long intervals, supplemented by a standard analytical toolbox.