System of imprimitivity

The concept of a system of imprimitivity is used in mathematics, particularly in algebra and analysis, both within the context of the theory of group representations.

It was used by George Mackey as the basis for his theory of induced unitary representations of locally compact groups.

The structure, combinatorial in this case, respected by translation shows that either K is a maximal subgroup of G, or there is a system of imprimitivity (roughly, a lack of full "mixing").

Mackey also used the idea for his explication of quantization theory based on preservation of relativity groups acting on configuration space.

This generalized work of Eugene Wigner and others and is often considered to be one of the pioneering ideas in canonical quantization.

Let G be a finite group and U a representation of G on a finite-dimensional complex vector space H. The action of G on elements of H induces an action of G on the vector subspaces W of H in this way: Let X be a set of subspaces of H such that Then (U,X) is a system of imprimitivity for G. Two assertions must hold in the definition above: holds only when all the coefficients cW are zero.

To generalize the finite dimensional definition given in the preceding section, a suitable replacement for the set X of vector subspaces of H which is permuted by the representation U is needed.

As it turns out, a naïve approach based on subspaces of H will not work; for example the translation representation of R on L2(R) has no system of imprimitivity in this sense.

A system of imprimitivity based on (G, X) consists of a separable Hilbert space H and a pair consisting of which satisfy Let X be a standard G space and μ a σ-finite countably additive invariant measure on X.

More generally, if the action of G on X is ergodic (meaning that X cannot be reduced by invariant proper Borel sets of X) then any system of imprimitivity on X is homogeneous.

We now discuss how the structure of homogeneous systems of imprimitivity can be expressed in a form which generalizes the Koopman representation given in the example above.

If Φ is a strict unitary cocycle then the restriction of Φ to the fixed point subgroup Gx is a Borel measurable unitary representation U of Gx on H (Here U(H) has the strong operator topology).

Then w acts on the dual of R2 by multiplication by the transpose matrix This allows us to completely determine the orbits and the representation theory.

These are parametrized by the set consisting of We can write down explicit formulas for these representations by describing the restrictions to N and H. Case 1.

The corresponding representation π is of the form: It acts on L2(R) with respect to Lebesgue measure and Case 2.