Editing System of imprimitivity (section)

== 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 
:<math> [ h \cdot \chi](n) = \chi(h^{-1} n h). </math>

''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

:<math> g \mapsto g \cdot x_0. </math>

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 '''R'''<sup>4</sup>.

===Example: the Heisenberg group ===
The [[Heisenberg group]] is the group of 3 &times; 3 ''real'' matrices of the form:

:<math> \begin{bmatrix} 1 & x & z \\0 & 1 & y \\ 0 & 0 & 1 \end{bmatrix}. </math>

This group is the semi-direct product of

:<math> H = \bigg\{\begin{bmatrix} 1 & w & 0 \\0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}: w \in \mathbb{R} \bigg\} </math>

and the abelian normal subgroup

:<math> N = \bigg\{\begin{bmatrix} 1 & 0 & t \\0 & 1 & s \\ 0 & 0 & 1\end{bmatrix}: s,t \in \mathbb{R} \bigg\}. </math>

Denote the typical matrix in ''H'' by [''w''] and the typical one in ''N'' by [''s'',''t'']. Then

:<math> [w]^{-1} \begin{bmatrix}s \\ t \end{bmatrix} [w] = \begin{bmatrix}s \\ - w s + t \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ -w & 1 \end{bmatrix} \begin{bmatrix} s \\ t \end{bmatrix} </math>

''w'' acts on the dual of '''R'''<sup>2</sup> by multiplication by the transpose matrix

:<math> \begin{bmatrix} 1 & -w \\ 0 & 1 \end{bmatrix}. </math>

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

''Orbit structure'': The orbits fall into two classes: 
*A horizontal line which intersects the ''y''-axis at a non-zero value ''y''<sub>0</sub>. In this case, we can take the quasi-invariant measure on this line to be Lebesgue measure.
* A single point (''x''<sub>0</sub>,0) on the ''x''-axis
[[Image:OrbitStructureDual.png|thumb|250px|Orbit structure on dual space]]
''Fixed point subgroups'': These also fall into two classes depending on the orbit:
* The trivial subgroup {0}
* 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
* '''R''' &minus; {0}. These are infinite-dimensional.
* Pairs (''x''<sub>0</sub>, λ) ∈ '''R''' &times; '''R'''. ''x''<sub>0</sub> 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 ''L''<sup>2</sup>('''R''') with respect to Lebesgue measure and 
:<math> (\pi [s,t] \psi)(x) = e^{i t y_0} e^{i s x} \psi (x). \quad </math>
:<math> (\pi[w] \psi)(x) = \psi(x+w y_0).\quad </math>

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

:<math> \pi [s,t] = e^{i s x_0}. \quad </math>

:<math> \pi[w] = e^{i \lambda w}. \quad </math>