4.2 Extension of CCR by a Quadratic Generator
59
4.2.3 In this section we shall restrict our attention to the cases of representations
U (G) obtained from U (G n ) by addition of only one generator A := A m of the form
(4.2.1) in the manner described above. Let h be any nonzero real symmetric matrix
with elements h jk ( j, k = 1, 2, . . . 2n) and let
A :=
1
2
h
jk X j X k
(4.2.8)
denote here the selfadjoint operator corresponding to the right hand side of (4.2.8).
According to (4.2.6) the operators X j ( j = 1, 2, . . . 2n), A and I H are selfadjoint
generators of an irreducible unitary representation U (G) of a 2n + 2—dimensional
connected Lie group G containing G n as a normal subgroup. The restriction U (G n )
of U (G) is irreducible, too. Let U(G) be the realization of G in P(H) obtained by
the natural projection of U (G), 2.3.9. We shall investigate infinitely differentiable
orbits of U(G) in P(H).
4.2.4 Lemma. Let U (G) be as in 4.2.3 and D G be a dense invariant subset in
H consisting of infinitely differentiable vectors of U (G), e.g. D G is the Gårding
subspace for U (G)([13, 11.1.8]). Let ϕ ∈ D G , ϕ = 1 and O ϕ := U(G)ϕ be the
immersed submanifold of P(H) according to 3.1.2. The orbit O ϕ is 2n-dimensional
iff there is an element C ϕ ∈ U (g),
C ϕ := c
j
ϕ X j − A,
(4.2.9)
such, that ϕ is its eigenvector:
C ϕ ϕ = λϕ f or some λ ∈ R.
(4.2.10)
If (4.2.9) with (4.2.10) is the case, then O ϕ = U(G n )ϕ.
Proof. The tangent space to O ϕ at ϕ is the linear hull of vectors σ j (ϕ) ( j =
1, 2, . . . 2n) (see 4.1.5) and σ A (ϕ) (see e.g. 2.3.5 and Sect. 3.2). According to 4.1.5,
all the σ j ’s are linearly independent. Hence O ϕ is 2n-dimensional iff
σ A (ϕ) = c
j
ϕ σ j (ϕ)
(4.2.11)
for some reals c
j
ϕ . According to (2.3.8), the equation (4.2.10) implies (4.2.11).
Assuming (4.2.11), we have in the standard identification of T ϕ P(H) with [ϕ]
⊥ by
the help of ϕ (see 2.1.7 and (2.1.16)):
(I − P ϕ )(c
j
ϕ X j − A)ϕ = 0 ⇒ (c
j
ϕ X j − A)ϕ = λϕ
(4.2.12)
with λ := λ(ϕ) := T r(P ϕ (c
j
ϕ X j − A)).
59
4.2.3 In this section we shall restrict our attention to the cases of representations
U (G) obtained from U (G n ) by addition of only one generator A := A m of the form
(4.2.1) in the manner described above. Let h be any nonzero real symmetric matrix
with elements h jk ( j, k = 1, 2, . . . 2n) and let
A :=
1
2
h
jk X j X k
(4.2.8)
denote here the selfadjoint operator corresponding to the right hand side of (4.2.8).
According to (4.2.6) the operators X j ( j = 1, 2, . . . 2n), A and I H are selfadjoint
generators of an irreducible unitary representation U (G) of a 2n + 2—dimensional
connected Lie group G containing G n as a normal subgroup. The restriction U (G n )
of U (G) is irreducible, too. Let U(G) be the realization of G in P(H) obtained by
the natural projection of U (G), 2.3.9. We shall investigate infinitely differentiable
orbits of U(G) in P(H).
4.2.4 Lemma. Let U (G) be as in 4.2.3 and D G be a dense invariant subset in
H consisting of infinitely differentiable vectors of U (G), e.g. D G is the Gårding
subspace for U (G)([13, 11.1.8]). Let ϕ ∈ D G , ϕ = 1 and O ϕ := U(G)ϕ be the
immersed submanifold of P(H) according to 3.1.2. The orbit O ϕ is 2n-dimensional
iff there is an element C ϕ ∈ U (g),
C ϕ := c
j
ϕ X j − A,
(4.2.9)
such, that ϕ is its eigenvector:
C ϕ ϕ = λϕ f or some λ ∈ R.
(4.2.10)
If (4.2.9) with (4.2.10) is the case, then O ϕ = U(G n )ϕ.
Proof. The tangent space to O ϕ at ϕ is the linear hull of vectors σ j (ϕ) ( j =
1, 2, . . . 2n) (see 4.1.5) and σ A (ϕ) (see e.g. 2.3.5 and Sect. 3.2). According to 4.1.5,
all the σ j ’s are linearly independent. Hence O ϕ is 2n-dimensional iff
σ A (ϕ) = c
j
ϕ σ j (ϕ)
(4.2.11)
for some reals c
j
ϕ . According to (2.3.8), the equation (4.2.10) implies (4.2.11).
Assuming (4.2.11), we have in the standard identification of T ϕ P(H) with [ϕ]
⊥ by
the help of ϕ (see 2.1.7 and (2.1.16)):
(I − P ϕ )(c
j
ϕ X j − A)ϕ = 0 ⇒ (c
j
ϕ X j − A)ϕ = λϕ
(4.2.12)
with λ := λ(ϕ) := T r(P ϕ (c
j
ϕ X j − A)).