page_title: System of imprimitivity
section_title: Example: the Heisenberg group
hierarchical_section_title: System of imprimitivity / Applications to the theory of group representations / Example: the Heisenberg group
context_page_description: 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 representations 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 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. 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.
context_section_description: The Heisenberg group is the group of 3 × 3 real matrices of the form:
  This group is the semi-direct product of
  and the abelian normal subgroup
  Denote the typical matrix in H by [w] and the typical one in N by [s,t]. Then
  w acts on the dual of R² by multiplication by the transpose matrix
  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 y₀. In this case, we can take the quasi-invariant measure on this line to be Lebesgue measure.
  A single point (x₀,0) on the x-axis.
  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 − {0}. These are infinite-dimensional.
  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 
  Case (2). The corresponding representation is given by the 1-dimensional character
caption: Orbit structure on dual space
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