  The non--mesonic weak decay of polarized $\Lambda$-hypernuclei is studied
with a microscopic diagrammatic formalism in which one- and two-nucleon induced
decay mechanisms, $\vec{\Lambda} N \to NN$ and $\vec{\Lambda} NN \to NNN$, are
considered together with (and on the same ground of) nucleon final state
interactions. We adopt a nuclear matter formalism extended to finite nuclei via
the local density approximation. Our approach adopts different
one-meson-exchange weak transition potentials, while the strong interaction
effects are accounted for by a Bonn nucleon-nucleon interaction. We also
consider the two-pion-exchange effect in the weak transition potential. Both
the two-nucleon induced decay mechanism and the final state interactions reduce
the magnitude of the asymmetry. The quantum interference terms considered in
the present microscopic approach give rise to an opposite behavior of the
asymmetry with increasing energy cuts to that observed in models describing the
nucleon final state interactions semi-classically via the intranuclear cascade
code. Our results for the asymmetry parameter in $^{12}_{\Lambda}$C obtained
with different potential models are consistent with the asymmetry measured at
KEK.
