  The $\Lambda(1405)$ resonance production reaction is investigated within the
framework of the coupled-channels Alt-Grassberger-Sandhas (AGS) equations. We
perform full three-body calculations for the $\bar{K}NN-\pi YN$ amplitudes on
the physical real energy axis and investigate how the signature of the
$\Lambda(1405)$ appears in the cross sections of the $K^-d\rightarrow \pi\Sigma
n$ reactions, also in view of the planned E31 experiment at J-PARC. Two types
of meson-baryon interaction models are considered: an energy-dependent
interaction based on chiral $SU(3)$ effective field theory, and an
energy-independent version that has been used repeatedly in phenomenological
approaches. These two models have different off-shell properties that imply
correspondingly different behavior in the three-body system. We investigate how
these features show up in differential cross sections of $K^- d\rightarrow
\pi\Sigma n$ reactions. Characteristic patterns distinguishing between the two
models are found in the invariant mass spectrum of the final $\pi\Sigma$ state.
The $K^-d\rightarrow \pi\Sigma n$ reaction, with different
($\pi^{\pm}\Sigma^{\mp}$ and $\pi^{0}\Sigma^{0}$) charge combinations in the
final state, is thus demonstrated to be a useful tool for investigating the
subthreshold behavior of the $\bar{K}N$ interaction.
