Wigner D-matrix
$$\begin{align}
R(\alpha, \beta, \gamma) = e^{-\mathrm{i}\alpha J_{z}}
e^{-\mathrm{i}\beta J_{y}}
e^{-\mathrm{i}g J_{z}}
\end{align}$$
$$\begin{align}
D^l_{m' m} (\alpha, \beta, \gamma) \langle l, m'| R(\alpha, \beta, \gamma) |l, m\rangle
\end{align}$$
$D^l_{m' m}(R)$ 是旋转操作 $R$ 在一组基底 $|l, m\rangle$ 下的表示的矩阵元( $l$ 是固定
的, 也就是说 $1=\sum_m |l, m\rangle\langle l, m|$ ).
$D^l(R)$ 是 $SO(3)$ 群元 $R$ 在不可约表示 $l$ 中的表示矩阵.
Relation to spherical harmonics
$$\begin{align}
Y_{l, m}^{*} (\theta, \phi) = \langle l, m | \theta, \phi \rangle
=& \langle l, m | R(\phi, \theta, \gamma) | \theta = 0, \phi = 0 \rangle \\
=&\sum_{m'} \langle l, m | R(\phi, \theta, \gamma) |l, m'\rangle\langle l, m' | \theta = 0, \phi = 0 \rangle \\
=&\sum_{m'} \langle l, m | R(\phi, \theta, \gamma) |l, m'\rangle \delta_{m', 0}\sqrt{\frac{2l + 1}{4\pi}} \\
=&\langle l, m | R(\phi, \theta, \gamma) |l, 0\rangle \sqrt{\frac{2l + 1}{4\pi}} \\
=& D^l_{m, 0}(\phi, \theta, \gamma) \sqrt{\frac{2l + 1}{4\pi}}
\end{align}$$
Reference
