  We present the dictionary between the one-particle Hilbert spaces of totally
symmetric tensor-spinor fields of spin $s={3}/{2}, {5}/{2}$ with any mass
parameter on $D$-dimensional ($D \geq 3$) de Sitter space ($dS_{D}$) and
Unitary Irreducible Representations (UIR's) of the de Sitter algebra
spin$(D,1)$. Our approach is based on expressing the eigenmodes on global
$dS_{D}$ in terms of eigenmodes of the Dirac operator on the ${(D-1)}$-sphere,
which provides a natural way to identify the corresponding representations with
known UIR's under the decomposition spin$(D,1)$ $\supset$ spin$(D)$.
Remarkably, we find that four-dimensional de Sitter space plays a distinguished
role in the case of the gauge-invariant theories. In particular, the strictly
massless spin-3/2 field, as well as the strictly and partially massless
spin-5/2 fields on $dS_{D}$, are not unitary unless $D=4$.
