  We study the relations between Dirac fields living on the 2-dimensional
Lorentzian cylinder and the ones living on the double-covering of the
2-dimensional de Sitter manifold, here identified as a certain coset space of
the group $SL(2,R)$. We show that there is an extended notion of de Sitter
covariance only for Dirac fields having the Neveu-Schwarz anti-periodicity and
construct the relevant cocycle. Finally, we show that the de Sitter symmetry is
naturally inherited by the Neveu-Schwarz massless Dirac field on the cylinder.
