System of imprimitivity

To motivate the general definitions, we first formulate a definition in the case of finite groups and their representations on finite-dimensional vector spaces.

Suppose G is 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 an obvious way:

${\displaystyle U_{g}W=\{U_{g}w:w\in W\}.}$

Suppose X is a set of subspaces of H such that

1. the elements of X are permuted by the action of G on subspaces and
2. H is the (internal) algebraic direct sum of the elements of X, i.e.,

${\displaystyle H=\bigoplus _{W\in X}W.}$

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

Two assertions must hold in the definition above:

1. the spaces W for W ∈ X must span H, and
2. the spaces W ∈ X must be linearly independent, that is,

${\displaystyle \sum _{W\in X}c_{W}v_{W}=0,\quad v_{W}\in W\setminus \{0\}}$

holds only 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 if and only if 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.

${\displaystyle G_{0}=\{g\in G:U_{g}W_{0}\subseteq W_{0}\}.}$

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 of a representation U defined by

${\displaystyle \chi _{U}(g)=\operatorname {tr} (U_{g}).}$

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

${\displaystyle \chi _{U}(g)={\frac {1}{|G_{0}|}}\sum _{\{x\in G:{x}^{-1}\,g\,x\in G_{0}\}}\chi _{V}({x}^{-1}\ g\ x),\quad \forall g\in G.}$

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

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. 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

${\displaystyle G\times X\rightarrow X,\quad (g,x)\mapsto g\cdot x.}$

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

${\displaystyle U_{g}\pi (A)U_{g^{-1}}=\pi (g\cdot A).}$

A system of imprimitivity is homogeneous of multiplicity n, where 1 ≤ n ≤ ω if and only if 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

${\displaystyle s(g,x)={\bigg [}{\frac {d\mu }{dg^{-1}\mu }}{\bigg ]}(x)\in [0,\infty )}$

is well-defined for every g ∈ G, where

${\displaystyle g^{-1}\mu (A)=\mu (gA).\quad }$

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

${\displaystyle s:G\times X\rightarrow [0,\infty )}$

is Borel measurable and satisfies

${\displaystyle s(g,x)={\bigg [}{\frac {d\mu }{dg^{-1}\mu }}{\bigg ]}(x)\in [0,\infty )}$

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

${\displaystyle \Phi :G\times X\rightarrow \operatorname {U} (H)}$

such that

${\displaystyle \Phi (e,x)=I\quad }$

for almost all x ∈ X

${\displaystyle \Phi (gh,x)=\Phi (g,h\cdot x)\Phi (h,x)}$

for almost all (g, h, x). A unitary cocycle is strict if and only if 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

${\displaystyle [U_{g}\psi ](x)={\sqrt {s(g,g^{-1}x)}}\ \Phi (g,g^{-1}x)\ \psi (g^{-1}x).}$

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

${\displaystyle \int _{X}^{\oplus }Hd\mu (x).}$

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

${\displaystyle \pi (A)\psi =1_{A}\psi ,\quad \int _{X}^{\oplus }Hd\mu (x)\rightarrow \int _{X}^{\oplus }Hd\mu (x),}$

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

Conversely, any homogeneous system of imprimitivity is of this form, for some measure σ-finite measure μ. This measure is unique up to measure equivalence, that is to say, two such measures have the same sets of measure 0.

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.

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

${\displaystyle G_{x}=\{g\in G:g\cdot x=x\}}$

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

${\displaystyle \Phi :G\times X\rightarrow \operatorname {U} (H)}$

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).

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

${\displaystyle [h\cdot \chi ](n)=\chi (h^{-1}nh).}$

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

${\displaystyle g\mapsto g\cdot x_{0}.}$

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.

Example: the Heisenberg group

The Heisenberg group is the group of 3 × 3 real matrices of the form:

${\displaystyle {\begin{bmatrix}1&x&z\\0&1&y\\0&0&1\end{bmatrix}}.}$

This group is the semi-direct product of

${\displaystyle H={\bigg \{}{\begin{bmatrix}1&w&0\\0&1&0\\0&0&1\end{bmatrix}}:w\in \mathbb {R} {\bigg \}}}$

and the abelian normal subgroup

${\displaystyle N={\bigg \{}{\begin{bmatrix}1&0&t\\0&1&s\\0&0&1\end{bmatrix}}:s,t\in \mathbb {R} {\bigg \}}.}$

Denote the typical matrix in H by [w] and the typical one in N by [s,t]. Then

${\displaystyle [w]^{-1}{\begin{bmatrix}s\\t\end{bmatrix}}[w]={\begin{bmatrix}s\\-ws+t\end{bmatrix}}={\begin{bmatrix}1&0\\-w&1\end{bmatrix}}{\begin{bmatrix}s\\t\end{bmatrix}}}$

w acts on the dual of R² by multiplication by the transpose matrix

${\displaystyle {\begin{bmatrix}1&-w\\0&1\end{bmatrix}}.}$

This allows us to completely determine the orbits and the representation theory.

Orbit structure: The orbits fall into two classes:

1. A horizontal line which intersects the y-axis at a non-zero value y₀. In this case, we can take the quasi-invariant measure on this line to be Lebesgue measure.
2. A single point (x₀,0) on the x-axis.

Orbit structure on dual space

Fixed point subgroups: These also fall into two classes depending on the orbit:

1. The trivial subgroup {0}.
2. The group H itself.

Classification: This allows us to completely classify all irreducible representations of the Heisenberg group. These are parametrized by the set consisting of

1. R − {0}. These are infinite-dimensional.
2. Pairs (x₀, λ) ∈ R × R. x₀ is the abscissa of the single point orbit on the x-axis and λ is an element of the dual of H These are one-dimensional.

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

${\displaystyle (\pi [s,t]\psi )(x)=e^{ity_{0}}e^{isx}\psi (x).\quad }$

${\displaystyle (\pi [w]\psi )(x)=\psi (x+wy_{0}).\quad }$

Case (2). The corresponding representation is given by the 1-dimensional character

${\displaystyle \pi [s,t]=e^{isx_{0}}.\quad }$

${\displaystyle \pi [w]=e^{i\lambda w}.\quad }$

• G. W. Mackey, The Theory of Unitary Group Representations, University of Chicago Press, 1976.
• V. S. Varadarajan, Geometry of Quantum Theory, Springer-Verlag, 1985.
• David Edwards, The Mathematical Foundations of Quantum Mechanics, Synthese, Volume 42, Number 1/September, 1979, pp. 1–70.