System of imprimitivity

The concept of 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 representationss of locally compact groups.

The simplest case, and the context in which the idea was first noticed, is that of finite groups. Consider a group G and subgroups H and K, with K contained in H. Then the left cosets of H in G are each the union of left cosets of K. Not only that, but translation (on one side) by any element g of G respects this decomposition. The connection with induced representations is that the permutation representation on cosets is the special case of induced representation, in which a representation is induced from a trivial representation. 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'). In order to generalise this to other cases, the concept is re-expressed: first in terms of functions on G constant on K-cosets, and then in terms of projection operators (for example the averaging over K-cosets of elements of the group algebra).

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.

Illustrative example

To motivate the general definitions, we first consider the case of finite groups and representations of these on finite dimensional vector spaces.

Suppose G is a finite group and U is a representation of G on a finite-dimensional complex vector space H. If X is a set of subspaces of H such that H is the (internal) algebraic direct sum of the spaces in X, written

Then (U, X) is a system of imprimitivity for G.

Two assertions must hold in the above definition:

The spaces W for W ∈ X must span H and

the spaces W ∈ X must be linearly independent, that is the only linear relation

holds when all the coefficients c_W are zero.

If the action of G on the elements of X is transitive, then we say this is a transitive system of imprimitivity.

Suppose G is a finite group, G₀ a subgroup of G. A representation U of G is induced from a representation V of G₀ iff there exist the following:

A transitive system of imprimitivity (U, X) and

A subspace W₀ ∈ X
such that G₀ is the fixed point subgroup of W under the action of G i.e,

and V is equivalent to the representation of G₀
on W₀ given by U_h | W₀ for h ∈ G₀. Note that by this definition, induced by is a relation between representations. We would like to show that there is actually a mapping on representations which corresponds to this relation.

For finite groups one can easily show that a well-defined inducing construction exists on equivalence of representations by considering the character function of a representation U defined by

In fact if a representation U of G is induced from a representation V of G₀, then

Thus the character function χ_U (and therefore U itself) is completely determined by χ_V.

Example

Let G be a finite group and consider the space H of complex-valued functions on G. The left regular representation of G on H is defined by

Now H can be considered as the algebraic direct sum of the one dimensional spaces W_x, for x ∈ G, where

The spaces W_x are permuted by L_g.

Infinite dimensional systems of imprimitivity

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 naīve approach base 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. The right formulation of direct sum decomposition is formulated in terms of projection-valued measures.

Mackey's original formulation was expressed in terms of a locally compact second countable (lcsc) group G, a standard Borel space X and a Borel group action

We will refer to this as a standard Borel G-space.

The definitions can be given in a much more general context, but the original setup used by Mackey is still quite general and requires fewer technicalities.

Definition. Let G be a lcsc group acting on a standard Borel space X. A system of imprimitivity based on (G, X) consists of a separable Hilbert space H and a pair consisting of

A strongly-continuous unitary representation U: g → U_g of G on H.

A projection-valued measure π on the Borel sets of X with values in the projections of H;
which satisfy

Example

Let X be a standard G space and μ a σ-finite countably additive invariant measure on X. This means

for all g ∈ G and Borel subsets A of G.

Let π(A) be multiplication by the indicator function of A and U_g be the operator

Then (U, π) is a system of imprimitivity of (G, X) on L²_μ(X).

This system of imprimitivity is sometimes called the Koopman system of imprimitivity.

Homogeneous systems of imprimitivity

A system of imprimitivity is homogeneous of multiplicity n, where 1 ≤ n ≤ ω iff the corresponding projection-valued measure π on X is homogeneous of multiplicity n. In fact, X breaks up into a countable disjoint family {X_n}_{1 ≤ n ≤ ω} of Borel sets such that π is homogeneous of multiplicity n on X_n. It is also easy to show X_n is G invariant.

Lemma. Any system of imprimitivity is an orthogonal direct sum of homogeneous ones.

It can be shown that if the action of G on X is transitive, then any system of imprimitivity on X is homogeneous. 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.

In the following, we assume that μ is a σ-finite measure on a standard Borel G-space X such that the action of G respects the measure class of μ. This condition is weaker than invariance, but it suffices to construct a unitary translation operator similar to the Koopman operator in the example above. G respects the measure class of μ means that the Radon-Nikodym derivative

is well-defined for every g ∈ G, where

It can be shown that there is a version of s which is jointly Borel measurable, that is

is Borel measurable and satisfies

for almost all values of (g, x) ∈ G × X.

Suppose H is a separable Hilbert space, U(H) the unitary operators on H. A unitary cocycle is a Borel mapping

such that

for almost all x ∈ X

for almost all (g, h, x). A unitary cocyle is strict iff the above relations hold for all (g, h, x). It can be shown that for any unitary cocycle there is a strict unitary cocycle which is equal almost everywhere to it (Varadarajan, 1985).

Theorem. Define

Then U is a unitary representation of G on the Hilbert space

Moreover, if for any Borel set A π is the projection operator

then (U, π) is a system of imprimitivity of (G,X).

Conversely, any system of imprimitivity is of this form.

Indeed much more can be said about the correspondence between homogeneous systems of imprimitivity and cocycles.

When the action of G on X is transitive however, the correspondence takes a particularly explicit form based on the representation obtained by restricting the cocycle Φ to a fixed point subgroup of the action. We consider this case in the next section.

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 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 iff 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).

Applications to the theory of group representations

Systems of imprimitivity arise naturally in the determination of the representations of a semi-direct product of an abelian group V by a group G of automorphisms of V. An important example of this is the inhomogeneous Lorentz group.

References

G. W. Mackey, The Theory of Unitary Group Representations, University of Chicago Press, 1976.
V. S. Varadarajan, Geometry of Quantum Theory, Springer-Verlag, 1985.

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