  We investigate the spin-dependent (naive) T-odd fragmentation function
$D^\perp_{1T}$, which can provide an explanation on the transverse polarization
of the $\Lambda^0$ hyperon produced in an unpolarized process. We calculate
$D^\perp_{1T}$ for light flavors in the spectator diquark model, with a
Gaussian form factor at the hyperon-quark-diquark vertex. We include in the
calculation both the scalar diquark and axial-vector diquark spectators. We
determine the values of the model parameters by fitting the unpolarized
fragmentation function $D^\Lambda_1$ to the DSV parametrization for
$D^\Lambda_1$. In addition, we compute the longitudinal polarization
fragmentation function $G^\Lambda_1$ and compare it with the known
parametrization of $G^\Lambda_1$. We also estimate the transverse polarizations
of $\Lambda$ production, in both semi-inclusive deep inelastic scattering and
single inclusive $e^+e^-$ annihilation.
