Analogs of Clifford’s theorem for polycyclic-by-finite groups

Abstract:

Let P be a primitive ideal in the group algebra $K[G]$ of the polycyclic group G and let N be a normal subgroup of G. We show that among the irreducible right $K[G]$-modules with annihilator P there exists at least one, V, such that the restricted $K[N]$-module ${V_N}$ is completely reducible, a sum of G-conjugate simple $K[N]$-submodules. Various stronger versions of this result are obtained. We also consider the action of G on the factor $K[N]/P \cap K[N]$ and show that, in case K is uncountable, any ideal I of $K[N]$ satisfying ${ \cap _{g \in G}}{I^g} = P \cap K[N]$ is contained in a primitive ideal Q of $K[N]$ with ${ \cap _{g \in G}}{I^g} = P \cap K[N]$.