  The purpose of this paper is to prove necessary and sufficient criteria for a
$GL(m|n)$-supermodule to have a good or Weryl filtration. We also introduce the
notion of a Steinberg supermodule analogous to the classical notion of
Steinberg module. We prove that the Steinberg supermodule inherits some
properties of the Steinberg module. Some new series of finite-dimensional
tilting supermodules are found.
