FIW-modules and stability criteria for representations of classical Weyl groups

We analyze the algebraic structure of the category of FIW-modules, and introduce restriction and induction operations that enable us to study interactions between the three families of groups. We use this theory to prove analogues of Murnaghan's 1938 stability theorem for Kronecker coefficients for the families Bn and Dn. The theory of FIW-modules gives a conceptual framework for stability results such as these.