The generalization of the "spherical tensor harmonics" of quantum mechanics to a general case of a compact Lie group is given as follows: Let $G$ be a compact Lie group and $H$ be a closed subgroup, then, the Hilbert space of square integrable functions on $G/H$ (which may be taken as the eigenfunctions of the Killing Laplacian) is a direct sum of $G$-representations called "spherical representations". These representations are characterized by having an $H$ single. Please see, for example, appendix B of HARMONIC ANALYSIS AND PROPAGATORS ON HOMOGENEOUS SPACES by Camporesi. This definition generalizes (and was named after) the spherical harmonics of quantum mechanics. In this case $G=SU(2)$, $H=U(1)$ and $G/H = SU(2)/U(1)$. The sphericality condition implies that the spherical harmonics can only be representations containing a $U(1)$ singlet, thus must be of integer spin.

The generalization of the "spherical tensor harmonics" of quantum mechanics to a general case of a compact Lie group is given as follows: Let $G$ be a compact Lie group and $H$ be a closed subgroup, then, the Hilbert space of square integrable functions on $G/H$ (which may be taken as the eigenfunctions of the Killing Laplacian) is a direct sum of $G$-representations called "spherical representations". These representations are characterized by having an $H$ singlet. Please see, for example, appendix B of HARMONIC ANALYSIS AND PROPAGATORS ON HOMOGENEOUS SPACES by Camporesi. This definition generalizes (and was named after) the spherical harmonics of quantum mechanics. In this case $G=SU(2)$, $H=U(1)$ and $G/H = SU(2)/U(1)$. The sphericality condition implies that the spherical harmonics can only be representations containing a $U(1)$ singlet, thus must be of integer spin.