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?
