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,


Bessel functions, Hermite polynomials, Legendre polynomials - do each of these represent some Lie group?  If so, what is it in each case?
Is there a nice example of some complete functions that represent a non-compact Lie group?
Is there a nice example of some complete functions that do not represent any Lie group at all?


Thanks!
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 $m - \ell \in \mathbb{Z}$.

The matrix elements of Wigner's small matrix give the Legendre polynomials:

$$ \langle \ell, 0 | e^{-i \theta J_z }|\ell, 0 \rangle = P_\ell(\cos \theta)$$  

Here $J_z$ is a generator of the $\mathfrak{so}(3)$ Lie Algebra and the exponent is a rotation , $e^{-i \theta J_z } \in SO(3)$.
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 $m - \ell \in \mathbb{Z}$.

The matrix elements of Wigner's small matrix give the Legendre polynomials:

$$ \langle \ell, 0 | e^{-i \theta J_z }|\ell, 0 \rangle = P_\ell(\cos \theta)$$  

Here $J_z$ is a generator of the $\mathfrak{so}(3)$ Lie Algebra and the exponent is a rotation , $e^{-i \theta J_z } \in SO(3)$.