# Special functions as representations of Lie Groups

-The spherical harmonics $Y_{lm}$ are complete on $L^2(S^2)$. They are also a representation of the (compact) Lie group $SO_3 (\mathbf{R})$.

-The functions $e^{i n x}$ are complete on $L^2([0,2\pi])$. They are also a representation of the (compact) Lie group $U(1).$

My question is basically, how general is this phenomenon? More specifically,

1. Bessel functions, Hermite polynomials, Legendre polynomials - do each of these represent some Lie group? If so, what is it in each case?

2. Is there a nice example of some complete functions that represent a non-compact Lie group?

3. Is there a nice example of some complete functions that do not represent any Lie group at all?

Thanks!

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A. Klimyk and N. Vilenkin, Studied this in their 3 volume monography. They also cover $SL(2, \mathbb{R})$ and $SU(1,1)$ which are not compact. – Sasha Nov 3 '11 at 23:19
Thank you for the suggestion. I looked at Klimyk and Vilenkin, and I'm sure that the answers to my questions are in there somewhere. However, these books are too advanced for me and I had trouble finding the answers to my questions. Can anyone help out with a more specific answer, or a more basic reference? – marlow Nov 11 '11 at 2:46
en.wikipedia.org/wiki/… – cactus314 Jan 6 '15 at 20:25
I think you guys would find this question math.stackexchange.com/questions/1163032/… interesting – bolbteppa Feb 24 '15 at 10:31

The representations of $SO(3)$ are indexed by numbers $|\ell, m \rangle$ where $\ell \in \frac{1}{2}\mathbb{Z}$ and $|m| \leq \ell$ with $m - \ell \in \mathbb{Z}$.
$$\langle \ell, 0 | e^{-i \theta J_z }|\ell, 0 \rangle = P_\ell(\cos \theta)$$
Here $J_z$ is a generator of the $\mathfrak{so}(3)$ Lie Algebra and the exponent is a rotation , $e^{-i \theta J_z } \in SO(3)$.