Editing System of imprimitivity (section)

===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>