Unitary Representations of Lie Groups with Reflection Symmetry
Abstract
We consider the following class of unitary representations
π
of some (real) Lie group
G
which has a matched pair of symmetries described as follows: (i) Suppose
G
has a period-2 automorphism
τ
, and that the Hilbert space
H(π)
carries a unitary operator
J
such that
Jπ=(π∘τ)J
(i.e., selfsimilarity). (ii) An added symmetry is implied if
H(π)
further contains a closed subspace
K
0
having a certain order-covariance property, and satisfying the
K
0
-restricted positivity:
<v∣Jv>≥0
,
∀v∈
K
0
, where
<⋅∣⋅>
is the inner product in
H(π)
. From (i)--(ii), we get an induced dual representation of an associated dual group
G
c
. All three properties, selfsimilarity, order-covariance, and positivity, are satisfied in a natural context when
G
is semisimple and hermitean; but when
G
is the
(ax+b)
-group, or the Heisenberg group, positivity is incompatible with the other two axioms for the infinite-dimensional irreducible representations. We describe a class of
G
, containing the latter two, which admits a classification of the possible spaces
K
0
⊂H(π)
satisfying the axioms of selfsimilarity and order-covariance.