In this special basis, the spin-weighted spherical harmonics appear as actual functions, because the choice of a polar axis fixes the U(1) gauge ambiguity.
At a point x on the sphere, a positively oriented orthonormal basis of tangent vectors at x is a pair a, b of vectors such that where the first pair of equations states that a and b are tangent at x, the second pair states that a and b are unit vectors, the penultimate equation that a and b are orthogonal, and the final equation that (x, a, b) is a right-handed basis of R3.
Concretely, these are understood as functions f on C2\{0} satisfying the following homogeneity law under complex scaling This makes sense provided s is a half-integer.
The spin weight bundles B(s) are equipped with a differential operator ð (eth).
Just as conventional spherical harmonics are the eigenfunctions of the Laplace-Beltrami operator on the sphere, the spin-weight s harmonics are the eigensections for the Laplace-Beltrami operator acting on the bundles E(s) of spin-weight s functions.
The obvious recursion relation results from repeatedly applying the raising or lowering operators.