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

Dimitrios Chatzakos, Corentin Darreye, Ikuya Kaneko

Published 2023-11-23Version 1

We study the mean value and quantum variance for ${\mathrm{SL}_{n}({\mathbb Z})}$ degenerate maximal parabolic Eisenstein series. We first establish a mean value result for the inner products \begin{equation*} \mu_{t, n}(f) = \int_{\mathrm{SL}_{n}(\mathbb{Z}) \backslash \mathfrak{h}^{n}} f(z) \left|E_{(n-1, 1)} \left(z, \frac{1}{2}+it, \mathbf{1} \right) \right|^{2} d\mu(z) \end{equation*} with $f$ in a restricted space of incomplete parabolic Eisenstein series. Our estimate breaks the barrier with a polynomial power saving beyond the prediction of the Generalised Lindel\"{o}f Hypothesis. Moreover, we prove a quantum variance formula for $\mu_{t, n}(f)$. This paper builds on the work of Zhang (2019) on quantum unique ergodicity for $\mathrm{SL}_{n}({\mathbb{Z}})$ degenerate Eisenstein series as well as the work of Huang (2021) on quantum variance for $\mathrm{SL}_{2}({\mathbb{Z}})$ Eisenstein series.