System of Imprimitivity - Induced Representations

Induced Representations

If X is a Borel G space and x ∈ X, then the fixed point subgroup

is a closed subgroup of G. Since we are only assuming the action of G on X is Borel, this fact is non-trivial. To prove it, one can use the fact that a standard Borel G-space can be imbedded into a compact G-space in which the action is continuous.

Theorem. Suppose G acts on X transitively. Then there is a σ-finite quasi-invariant measure μ on X which is unique up to measure equivalence (that is any two such measures have the same sets of measure zero).

If Φ is a strict unitary cocycle

then the restriction of Φ to the fixed point subgroup G_x is a Borel measurable unitary representation U of G_x on H (Here U(H) has the strong operator topology). However, it is known that a Borel measurable unitary representation is equal almost everywhere (with respect to Haar measure) to a strongly continuous unitary representation. This restriction mapping sets up a fundamental correspondence:

Theorem. Suppose G acts on X transitively with quasi-invariant measure μ. There is a bijection from unitary equivalence classes of systems of imprimitivity of (G, X) and unitary equivalence classes of representation of G_x.

Moreover, this bijection preserves irreducibility, that is a system of imprimitivity of (G, X) is irreducible if and only if the corresponding representation of G_x is irreducible.

Given a representation V of G_x the corresponding representation of G is called the representation induced by V.

See Theorem 6.2 of (Varadarajan, 1985).