GL(m|n)GL(m|n)-supermodules with good and Weyl filtrations

The purpose of this paper is to prove necessary and sufficient criteria for a GL(m|n)GL(m|n)-supermodule to have a good or Weyl 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.