# Index position of spherical harmonic function and tensorial properties

The spherical harmonics functions are denoted as $$Y_l^m$$ or $$Y_{lm}$$in https://en.wikipedia.org/wiki/Spherical_harmonics

e.g., \begin{align} Y_{lm} &= \dfrac{i}{\sqrt{2}} \left(Y_\ell^{m} - (-1)^m\, Y_\ell^{-m}\right) \text{if}\ m < 0 \end{align}

My question is, does the super and lower indices related to tensor properties as covariant/contravariant? E.g., $$Y_{lm} = \sum_n g_{mn} Y_l^n$$. I may guess $$g_{mn} = \frac{i}{\sqrt{2}}, -(-1)^m \frac{i}{\sqrt{2}}, n=m \text{ or} -m; m<0$$. If yes, is there any advantage to utilize the tensor properties?

No. The distinction being made here is between real functions and complex functions. The Wikipedia $$Y_{lm}$$ are real functions forming an orthonormal basis on the sphere, while the $$Y_l^m$$ are a basis of complex functions. For example for $$l=1$$ the $$Y_{1m}$$ are $$\cos\theta$$, $$\sin\theta\cos\phi$$ and $$\sin\theta\sin\phi$$, while the $$Y_1^m$$ are $$\cos\theta$$, $$\sin\theta e^{\pm i\phi}$$.
Another significant distinction between them is in their eigenvalue properties. The $$Y_l^m$$ are eingenfunctions of $$L_z$$ with eigenvalues $$m$$, while the $$Y_{lm}$$ are eigenfunctions of $$L_x$$, $$L_y$$ and $$L_z$$ with eigenvalue zero.