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Thanks YCor for your comments. Is it possible to replace the group U(H) by something else that would contain scalar matrices ? I'm asking since the fact that the elements of U(H) are bounded operators on H is may be not so essential since afterall we give to U(H) the discrete topology.
Thanks @RW for the reference. I'll look at the paper but do you know if we can dig from their paper an aymptotic for the Green function of $X=\mathfrak{h}\times\mathfrak{h}$ as $r\rightarrow \infty$ where $r=\sqrt{r_1^2+r_2^2}$ stands for the hyperbolic radius on $X$ ?
