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Short question

Is the spectrum of a field generated by a Gaussian stationary process on a manifold with constant curvature diagonal if the field is expanded in terms of eigenfunctions of the covariant Laplacian?

Long question

In flat space, the eigenfunctions of the Laplace operator are (using Einstein summation convention) $\phi(\mathbf{x},\mathbf{k})=\exp (ix^\mu k_\mu$). A field $f(\mathbf{x})$ can be expanded as $$ f(\mathbf{x})=\int d^3\mathbf{k}\,\phi(\mathbf{x},\mathbf{k})\tilde{f}(\mathbf{k}). $$ Under a translation $\mathbf{x}\rightarrow \mathbf{x}+\mathbf{a}$, the eigenfunctions transform as $\phi\rightarrow\exp(ia^\mu k_\mu)\phi$. Thus, the two-point correlation function $$ \underbrace{\langle f^*(\mathbf{x})f(\mathbf{y})\rangle}_{\equiv C(\mathbf{x},\mathbf{y})}=\int d^3\mathbf{k}d^3\mathbf{q}\,\phi^*(\mathbf{x},\mathbf{k})\phi(\mathbf{y},\mathbf{q})\underbrace{\langle \tilde{f}^*(\mathbf{k})\tilde{f}(\mathbf{q})\rangle}_{\equiv \tilde{C}(\mathbf{k},\mathbf{q})} $$ transforms as $$ C(\mathbf{x},\mathbf{y})\rightarrow\int d^3\mathbf{k}d^3\mathbf{q}\,\phi^*(\mathbf{x},\mathbf{k})\phi(\mathbf{y},\mathbf{q})\tilde{C}(\mathbf{k},\mathbf{q})\times\exp(ia^\mu(k_\mu-q_\mu)) $$ and for the two-point function to be invariant under arbitrary translations, we require $\tilde{C}\propto\delta^3(\mathbf{k}-\mathbf{q})$. Note that $\langle\rangle$ denotes an average over realisations of the field.

However, in curved space, we cannot separate variables in Cartesian coordinates and express the eigenfunctions in terms of polar coordinates such that $\phi_{lm}(\mathbf{x},k)=Y_{lm}(\hat{\mathbf{x}})X_l(x,k)$, where $Y_lm$ are spherical harmonics and $X_l(x,k)$ are radial functions that reduce to spherical Bessel functions in the limit of flat space.

A field may be expanded as $$ f(\mathbf{x})=\int dk\,\sum_{lm}\phi_{lm}(\mathbf{x},k)f_{lm}(k) $$ Is the two-point correlation function of expansion coefficients still diagonal under generalised translations (fixed-point free isometries)? I.e. is $\langle f_{lm}^*(k) f_{l'm'}(k')\rangle\propto\delta(k-k')\delta_{mm'}\delta{ll'}$? If so, why? Does the spectrum only depend on $k$ or on $k$, $l$, and $m$ if the process is stationary under rotations (i.e. isometries with a fixed point)?

I have tried my luck with google but most texts are written for mathematicians; unfortunately I am a phycisist without too much formal mathematics background.

Help is much appreciated.

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