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-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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    I think you guys woul$d$ find this question http://m$a$th.stackexchange.com/questions/1163032/geometric-intuitive-meaning-of-sl2-r-su2-etc-representation-the interesting2015-02-24

1 Answers 1

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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 , $e^{-i \theta J_z } \in SO(3)$.