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While reading the appendix to 4th chapter of Iwaniec and Kowalski's analytic number theory I came upon a remark relating unitary representations and some integral transforms involving J-Bessel functions.

Let $g \in C_c^\infty(\mathbb{R}^+)$ and let $x$ denote a variable in $\mathbb{R}^k$. Also let $P$ be a spherical harmonic of degree $d$, meaning a degree $d$ homogeneous polynomial satisfying $\Delta P = 0$ with $\Delta$ the standard Laplacian.

Put $f(x) = P(x)g(|x|^2)|x|^{-\frac k2 +1}$. Then if $\hat{f}$ denotes the $k$-dimensional Fourier transform of $f$, one has $$\hat{f}(y) = P(y)h(|y|^2) |y|^{-\frac k2 +1}$$ where $$h(t) = \pi \int_0^\infty J_{\frac k2-1}(2\pi \sqrt{tu}) g(u) \mathrm{d} u.$$

Then the book goes on to remark that "These facts appear naturally in the context of unitary representations". I would be grateful if someone could explain what this integral formula has to do with unitary representations, and unitary representations of which group?

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Spherical harmonics naturally form a representation of $\text{O}(k)$. – Qiaochu Yuan Feb 10 '12 at 23:36
I don't have the book at hand, but one natural place to look would be N.J. Vilenkin, Special functions and the theory of group representations. – Alain Valette Feb 11 '12 at 6:59
I take it that we take the space of functions spanned by spherical harmonics, and act on a function on this space say Y via $g Y (x) = Y(gx)$ where $g\in O(k)$, and $x$ a point on the sphere. Why do we care that $Y$ is an eigenfunction of the Laplacian, what I have just described could as well be done with any function on the sphere. – Eren Mehmet Kiral Feb 13 '12 at 3:21

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