: ''This article addresses the notion of quasiregularity in the context of
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
and
topological algebra In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
Definition
A topological algebra A over a topological field K is ...
. For other notions of quasiregularity in
mathematics, see the disambiguation page
quasiregular.''
In mathematics, quasiregular representation is a concept of representation theory, for a
locally compact group
In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...
''G'' and a
homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ...
''G''/''H'' where ''H'' is a
closed subgroup
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
.
In line with the concepts of
regular representation
In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation.
One distinguishes the left regular r ...
and
induced representation
In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" represen ...
, ''G'' acts on functions on ''G''/''H''. If however
Haar measures give rise only to a
quasi-invariant measure
In mathematics, a quasi-invariant measure ''μ'' with respect to a transformation ''T'', from a measure space ''X'' to itself, is a measure which, roughly speaking, is multiplied by a numerical function of ''T''. An important class of examples o ...
on ''G''/''H'', certain 'correction factors' have to be made to the action on functions, for
:''L''
2(''G''/''H'')
to afford a
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ' ...
of ''G'' on
square-integrable function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute val ...
s. With appropriate scaling factors, therefore, introduced into the action of ''G'', this is the ''quasiregular representation'' or modified induced representation.
{{DEFAULTSORT:Quasiregular Representation
Unitary representation theory
Topological groups