A
(right)
regular representation of a group
is a linear representation
of
on a space
of complex-valued functions on
,
defined by the formula
provided that
separates the points of
and has the property that the function
,
,
belongs to
for all
,
.
Similarly, the formula
defines a (left) regular representation of
on
,
where the function
,
,
is assumed to belong to
for all
,
.
If
is a topological group, then
is often the space of continuous functions on
.
If
is locally compact, then the (right) regular representation of
is the (right) regular representation of
on the space
constructed by means of the right-invariant
Haar measure
on
;
the regular representation of a locally compact group is a continuous
unitary representation,
and the left and right regular representations are unitarily equivalent.