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I am currently looking for a formulation of a Fourier transform on manifolds of constant negative curvature. Specifically, I am looking for generalizations of the two dimensional results on the poincare disk parameterization given in [1].

Using notation $x = (x_1,x_2,x_3,\ldots)$, I have found that the metric of the poincare disk generalizes to a poincare ball , taking the points with $x_1^2+x_2^2+x_3^2+\ldots <1$ (e.g. $x\in B_1$), and using the metric $\frac{d x_1^2+dx_2^2+dx_3^2+\ldots}{1-|x|^2}$. For ease of notation I call the dimension of the poincare ball $d$.

I have already found that (some of) the eigenfunctions of the laplace-beltrami operator are given (from [1]) by functions of the form $e^{-\mu \tanh^{-1}\left(1+\frac{1-|x|^2}{1-x\cdot b}\right)}$, where $b=(b_1,b_2,b_3\ldots)\in S^d$ (e.g. $b_1^2+b_2^2+b_3^2+\ldots = 1$). The eigenvalues of these functions are given by $\mu(\mu-2d+2)$.

I then tried to define a forward fourier transform through $\bar{f}(\lambda,b) = \int_{B_1}f(x)e^{-(-i\lambda+d-1) \tanh^{-1}\left(1+\frac{1-|x|^2}{1-x\cdot b}\right)}$, mirroring the 2-dimensional result. However, I am unable to find a proper inverse transform. Based on symmetry in the formulation, my best assumption at the moment is that the inverse transform is of the form $\int_{\mathbb{R}}\int_{S^d}\bar{f}(\lambda,b)e^{-(i\lambda+d-1) \tanh^{-1}\left(1+\frac{1-|x|^2}{1-x\cdot b}\right)}g(\lambda)dbd\lambda$. Again, due to helgason[1] I already know that for the two dimensional case, $g(\lambda) = \frac{1}{4\pi}\lambda\tanh(\frac{\pi\lambda}{2})$, but I have been unable to find the proper generalization for this in higher dimensions. Concretely, my question is thus: how does $g(\lambda)$ generalize to higher dimensions? Alternatively, if someone could point me to a fourier type transform for some other parameterization of hyperbolic space in $d$ dimensions that would also be immensely useful.

I have so far tried to approach the problem by learning and applying techniques of harmonic analysis. Unfortunately, the introductory books I have found on the subject[2,3] do not seem to deal with the locally compact non-abelian case.

References:

  1. Sigurdur Helgason, Topics in Harmonic Analysis on Homogeneous Spaces
  2. Yitzhak Katznelson, An introduction to harmonic analysis
  3. Gerald B. Folland, A course in abstract harmonic analysis.
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  • $\begingroup$ There are a number of related SE questions/answers: math.stackexchange.com/questions/13902/… In general this is the subject of micro-local analysis, for which the literature is pretty large. Berger's book on Riemannian Geometry discusses the issues in chapter 9. A quick google search will turn up many recent books. $\endgroup$
    – JohnS
    May 1, 2018 at 14:47
  • $\begingroup$ I have taken a look at Berger's book, and the book by Buser (geometry and spectra of compact surfaces) it refers to, but these seem to limit themselves to the compact case with respect to surfaces of constant negative curvature. Is there currently known an explicit form of fourier transforms on the non-compact case? $\endgroup$
    – davidv1992
    May 3, 2018 at 11:43

1 Answer 1

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This may be explained in Helgason's Geometric Analysis on Symmetric Spaces. Theorem 1.3 on Page 201 lists a general Fourier inversion formula.

If I am parsing everything right, the function $g(\lambda)$ you are asking about is given in the formula as $|\mathbf{c}(\lambda)|^{-2}$, and where $\mathbf{c}$ is Harish-Chandra's $\mathbf{c}$-function.

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  • $\begingroup$ Yes, indeed, and, yes, indeed, it is not so easy to understand the general mechanism. Amazingly/luckily, one does not have to understand that Plancherel relation to understand the spectral decomposition of automorphic forms (though there are, indeed, other complications). :) $\endgroup$
    – paul garrett
    Jan 11, 2022 at 0:29

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