Symmetric tensor spherical harmonics on the N-sphere and their application to the de Sitter group SO(N,1)

The symmetric tensor spherical harmonics (STSH’s) on the N-sphere (SN), which are defined as the totally symmetric, traceless, and divergence-free tensor eigenfunctions of the Laplace–Beltrami (LB) operator on SN, are studied. Specifically, their construction is shown recursively starting from the lower-dimensional ones. The symmetric traceless tensors induced by STSH’s are introduced. These play a crucial role in the recursive construction of STSH’s. The normalization factors for STSH’s are determined by using their transformation properties under SO(N+1). Then the symmetric, traceless, and divergence-free tensor eigenfunctions of the LB operator in the N-dimensional de Sitter space-time which are obtained by the analytic continuation of the STSH’s on SN are studied. Specifically, the allowed eigenvalues of the LB operator under the restriction of unitarity are determined. Our analysis gives a group-theoretical explanation of the forbidden mass range observed earlier for the spin-2 field theory in de Sitter space-time.