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Let $E(z,1/2+it)$ be the Eisenstein series furnishing the continuous spectrum of the Laplace operator $\Delta$ on $X=PSL_2(\mathbb{Z})\setminus H^2$ and $dV(z)=y^{-2} \,dx \,dy$ be the volume element of the upper half plane $H^2$. In analogy with quantum mechanics, Luo-Sarnak defined the measure $\mu_t=|E(z,1/2+it)|^2 \,dV(z)$ and showed that it fulfills \begin{equation} \lim_{t\to\infty} \frac{\mu_t(K_1) }{\mu_t(K_2)}=\frac{\operatorname{Vol}(K_1)}{\operatorname{Vol}(K_2)} \end{equation} for compact, Jordan-measurable subsets $K_1,K_2$ of $X$. In analogy with the case of compact manifolds they called it quantum ergodicity. Another piece of work by Koyama, Sarnak and Petridis showed that this is also the case for certain arithmetic 3-manifolds (e.g. $X=PSL_2(\mathcal{O}_K)\setminus H^3$, where $\mathcal{O}_K$ is the integer ring of an imaginary quadratic field $K$ of class number one).

Now I am wondering how much is known for general arithmetic quotients of n-dimensional hyperbolic space or if the knowledge about these kind of examples ends with dimension 3. I am definitely grateful for any information on the current state of the matter and further references!

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    $\begingroup$ I think it's widely open for arithmetic quotients $n$-dimensional hyperbolic space $\mathbb{H}^n \cong \mathrm{SO}(n,1)/\mathrm{SO}(n)$, since we don't know how to relate the integrals of automorphic forms on these groups to $L$-functions except in very specific cases (the Gan-Gross-Prasad conjecture) that only apply to QUE when $n \in \{2,3\}$. $\endgroup$
    – Peter Humphries
    Commented Mar 30, 2020 at 14:13
  • $\begingroup$ Thanks for the insights, Peter! $\endgroup$
    – Claudius
    Commented Mar 31, 2020 at 15:21

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If $K$ is a real quadratic field of degree $n$ Truelsen (see https://arxiv.org/abs/0706.4239) showed QUE for for Eisenstein series on the arithmetic quotient $\text{PSL}_2(O_K)\backslash (H^2)^n$. I am not aware for a reference dealing with arbitrary number fields $K$ even if it should be possible to do this.

A true higher rank example has been worked out by L. Zhang (see https://arxiv.org/abs/1609.01386). Here the Eisenstein series on $\text{SL}_n(\mathbb{Z})\backslash \text{SL}_n(\mathbb{R})/\text{SO}(n,\mathbb{R})$ associated to a maximal parabolic subgroup are considered.

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  • $\begingroup$ Thank you for your fast reply and the references! Since I've read Zhang's paper I am curious whether there is a paper fully proving QE for SL3(Z)∖SL3(R)/SO(3,R) as there are explicit formulae available for coefficients of the Eisenstein series and the Hecke theory has already been worked out. Concerning Truelsen's paper I had the feeling that he is more or less dealing with n copies of PSL2(Z)∖H2 rather than an arithmetic quotient of a higher-dimensional hyperbolic space.But my understanding of his paper is limited. $\endgroup$
    – Claudius
    Commented Mar 30, 2020 at 13:23
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    $\begingroup$ Concerning the higher rank groups $\mathrm{SL}_n(\mathbb{Z})\backslash \mathrm{SL}_n(\mathbb{R}) / \mathrm{SO}(n)$, QUE is only known for certain degenerate Eisenstein series because these are the only class of automorphic forms on these higher groups for which an analogue of the Watson-Ichino formula holds. $\endgroup$
    – Peter Humphries
    Commented Mar 30, 2020 at 13:39

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