  Quasinormal modes of a massive Dirac field were calculated for various static
black hole backgrounds with the help of the WKB formula. These estimations,
however, are rough and valid only for very small values of $\mu M$, where $M$
and $\mu$ are mass of the black hole and field respectively. Thus, no accurate
calculations of massive Dirac modes are known even for the Schwarzschild black
hole and this is all the more so for the Kerr solution. Here we calculate
quasinormal modes of a massive Dirac field in the Kerr background. We have
shown that the infinitely long-lived quasinormal modes (quasiresonances), which
exist for boson fields, appear also in the fermions' quasinormal spectrum. Two
chiralities of massive fermions lead to an additional "fine structure" in the
spectrum. We discuss the effect of this fine structure on the behavior of
quasiresonances and the stability. The analysis is also extended to a charged
massive field in the Kerr-Newman background.
