Special functions as representations of Lie Groups

Question

-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!