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The divergence-free and gamma-traceless vector-spinor eigenfunctions, as wellas the divergence-free and gamma-traceless rank-2 symmetric tensor-spinoreigenfunctions, of the Dirac operator on the $N$-sphere ($S^{N}$) are writtendown explicitly for $N \geq 3$. The spin-3/2 and spin-5/2 eigenmodes of theDirac operator with arbitrary imaginary mass parameter on $N$-dimensional($N\geq 3$) de Sitter spacetime ($dS_{N}$) are obtained by analyticcontinuation. Their transformation properties under the de Sitter algebraspin$(N,1)$ are studied. For $N$ odd, we show that there is no de Sitter (dS)invariant scalar product for these eigenmodes. For $N$ even, although dSinvariant scalar products exist, positive-definiteness of the norm occurs onlyfor the strictly and partially massless theories in $N=4$ dimensions. For$N=4$, the way in which the eigenmodes form unitary strictly and partiallymassless representations of spin$(4,1)$ is emphasised. The analysis presentedin this paper reveals previously unknown features of the gauge-invarianttheories with spin 3/2 and 5/2 on $dS_{N}$ ($N \geq 3$): the strictly masslessspin-3/2 field theory, as well as the strictly and partially massless spin-5/2field theories, are unitary only for $N=4$. In particular, a unitary theory forthe gravitino field on $dS_{N}$ does not exist unless $N=4$.

The (partially) massless spin-3/2 and spin-5/2 fields in de Sitter spacetime as unitary and non-unitary representations of the de Sitter algebra