System of Imprimitivity - Applications To The Theory of Group Representations

Applications To The Theory of Group Representations

Systems of imprimitivity arise naturally in the determination of the representations of a group G which is the semi-direct product of an abelian group N by a group H that acts by automorphisms of N. This means N is a normal subgroup of G and H a subgroup of G such that G = N H and N ∩ H = {e} (with e being the identity element of G).

An important example of this is the inhomogeneous Lorentz group.

Fix G, H and N as above and let X be the character space of N. In particular, H acts on X by

Theorem. There is a bijection between unitary equivalence classes of representations of G and unitary equivalence classes of systems of imprimitivity based on (H, X). This correspondence preserves intertwining operators. In particular, a representation of G is irreducible if and only if the corresponding system of imprimitivity is irreducible.

This result is of particular interest when the action of H on X is such that every ergodic quasi-invariant measure on X is transitive. In that case, each such measure is the image of (a totally finite version) of Haar measure on X by the map

A necessary condition for this to be the case is that there is a countable set of H invariant Borel sets which separate the orbits of H. This is the case for instance for the action of the Lorentz group on the character space of R4.