136
P. Moylan
Now we introduce a ∗ structure on
D(p) which in view of Eqs. (3) and Theorem
2 induces a ∗ structure on
D(g) via the homomorphism τ λ . Let † be the adjoint
associated with this ∗ structure, then we have
Theorem 3 If L †
μ,ν = − L μ,ν , L
†
4,μ = − L 4,μ and if
Y † =
Y , then P μ =
D −1 A
ρ
μ L −1,ρ
and also P †
μ =
D −1 A μ,ρ L −1,ρ
† = L
†
−1,ρ
A μ,ρ†
D † −1 =
− P μ . Furthermore
ˆ
P μ , ˆ
P ν
= 0.
A proof of this theorem can be found in [5].
Although a representation of g always gives rise to a representation of the
enveloping algebra U (g), it does not necessarily give a representation of the skew
field. We have instead the following [5, 6]:
Theorem 4 Let (dπ, H) be an infinitesimally unitarizable representation of g on
an Hilbert space H, and let
Y be a self-adjoint operator on H which satisfies
Y 4 +
dπ( D
2 )
Y 2 +dπ( D
4 ) = 0. Then, if both dπ( D) −1 and dπ( D) † −1 exist on a suitable,
dense domain in H, there exists a skew symmetric representation d π of p on H
defined by: d π(L i,j ) = dπ( L i,j ), d π(P 0 ) = dπ( D) −1 dπ(
3
μ=0
A
μ
0
L −1,μ ), and
d π(P i ) = [d π(L i,0 ), d π(P 0 )](i = 1, 2, 3).
Note that it is necessary for both of the operators dπ( D) −1 and dπ( D) † −1 to exist
on the suitable dense domain in H postulated in this theorem for the existence of
mutually commuting translation operators d π(P μ ) on the representation space H of
the representation (dπ, H).
4 Representations
Let χ ν ∈ C be a complex character of A, i.e. χ ν (a(t)) = e (ν+3/2)t with ν ∈ C and
let (π (σ,,) , V (σ,,) ) be the representation π σ ⊗χ of M where χ ∈
{1, σ 3 } ( = 0, 1)
and π σ is a unitary representation of SL(2, R) on the complex vector space V (ρ,,) .
Consider π (σ,,) ⊗ χ ν : MA → V (σ,,) and extend this to a representation from P to
V (σ,,) by requiring that it act trivially on N.
Definition 1 Let G = Sp(2, R) and consider the space
H
(σ,,,ν)
:= I nd
G
P (π
(σ,,)
⊗ χ ν ⊗ 1 N ) =
f : G → V
(σ,,)
| f ∈ C
∞ (G)
f (gman) = π
(σ,,) (m)χ ν (a)f (g) for g ∈ G, m ∈ M, a ∈ A, n ∈ N
.
For f ∈ I nd G
P (π (σ,,) ⊗ χ ν ⊗ 1 N ) and g ∈ G we define a representation π (σ,,,ν) of
G on H (σ,,,ν) by π (σ,,,ν) (g)f (g ) = f (gg ) with g ∈ G.
P. Moylan
Now we introduce a ∗ structure on
D(p) which in view of Eqs. (3) and Theorem
2 induces a ∗ structure on
D(g) via the homomorphism τ λ . Let † be the adjoint
associated with this ∗ structure, then we have
Theorem 3 If L †
μ,ν = − L μ,ν , L
†
4,μ = − L 4,μ and if
Y † =
Y , then P μ =
D −1 A
ρ
μ L −1,ρ
and also P †
μ =
D −1 A μ,ρ L −1,ρ
† = L
†
−1,ρ
A μ,ρ†
D † −1 =
− P μ . Furthermore
ˆ
P μ , ˆ
P ν
= 0.
A proof of this theorem can be found in [5].
Although a representation of g always gives rise to a representation of the
enveloping algebra U (g), it does not necessarily give a representation of the skew
field. We have instead the following [5, 6]:
Theorem 4 Let (dπ, H) be an infinitesimally unitarizable representation of g on
an Hilbert space H, and let
Y be a self-adjoint operator on H which satisfies
Y 4 +
dπ( D
2 )
Y 2 +dπ( D
4 ) = 0. Then, if both dπ( D) −1 and dπ( D) † −1 exist on a suitable,
dense domain in H, there exists a skew symmetric representation d π of p on H
defined by: d π(L i,j ) = dπ( L i,j ), d π(P 0 ) = dπ( D) −1 dπ(
3
μ=0
A
μ
0
L −1,μ ), and
d π(P i ) = [d π(L i,0 ), d π(P 0 )](i = 1, 2, 3).
Note that it is necessary for both of the operators dπ( D) −1 and dπ( D) † −1 to exist
on the suitable dense domain in H postulated in this theorem for the existence of
mutually commuting translation operators d π(P μ ) on the representation space H of
the representation (dπ, H).
4 Representations
Let χ ν ∈ C be a complex character of A, i.e. χ ν (a(t)) = e (ν+3/2)t with ν ∈ C and
let (π (σ,,) , V (σ,,) ) be the representation π σ ⊗χ of M where χ ∈
{1, σ 3 } ( = 0, 1)
and π σ is a unitary representation of SL(2, R) on the complex vector space V (ρ,,) .
Consider π (σ,,) ⊗ χ ν : MA → V (σ,,) and extend this to a representation from P to
V (σ,,) by requiring that it act trivially on N.
Definition 1 Let G = Sp(2, R) and consider the space
H
(σ,,,ν)
:= I nd
G
P (π
(σ,,)
⊗ χ ν ⊗ 1 N ) =
f : G → V
(σ,,)
| f ∈ C
∞ (G)
f (gman) = π
(σ,,) (m)χ ν (a)f (g) for g ∈ G, m ∈ M, a ∈ A, n ∈ N
.
For f ∈ I nd G
P (π (σ,,) ⊗ χ ν ⊗ 1 N ) and g ∈ G we define a representation π (σ,,,ν) of
G on H (σ,,,ν) by π (σ,,,ν) (g)f (g ) = f (gg ) with g ∈ G.