4.3 Notes on Other Examples
69
O
◦
ϕ+ := (P
ϕ
+ )
−1
(0) ⊂
O ϕ
(4.3.20)
is closed in
O ϕ , and
O ϕ+ :=
O ϕ \
O
◦
ϕ+ is a submanifold of
O ϕ .
Each point of P
ϕ
+
O ϕ+ has a well defined projection into P(H
+
N ) and the mapping
P P
ϕ
+ ,
P P
ϕ
+ :
O ϕ+ → P(H
+
N ), ϕ
→ P P
ϕ
+ ϕ
:= {λP
ϕ
+ ϕ
: λ ∈ C} ∈ P(H
+
N ), (4.3.21)
is real analytic. The number rg(ϕ
) ∈ Z + (ϕ
∈
O ϕ+ ) :
rg(ϕ
) := rank T ϕ (P P
ϕ
+ ),
(4.3.22)
where T ϕ is the tangent mapping in an arbitrarily chosen point ϕ
∈
O ϕ+ , is given
in some charts on
O ϕ+ around ϕ
and on P(H
+
N ) around P P
ϕ
+ ϕ
as the dimension
of the vector space
5 T ϕ (P P
ϕ
+ )[T ϕ
O ϕ+ ] (which is, roughly speaking, the maximal
rank of submatrices of the mapping T ϕ (P P
ϕ
+ ) in these charts with nonvanishing
determinants). The function ϕ
→ rg(ϕ
) is lower semicontinuous, and possesses
only finite number of values. Hence for m ϕ := max{rg(ϕ
) : ϕ
∈
O ϕ+ } the subset
O
m
ϕ+ of
O ϕ defined by:
O
m
ϕ+ := rg
−1
(m ϕ ) := {ϕ
∈
O ϕ+ : rg(ϕ
) = m ϕ },
(4.3.23)
is open, hence it is a submanifold of
O ϕ . We can assume that ϕ was chosen such,
that ϕ ∈
O
m
ϕ+ . Let, for any ψ ∈
O
m
ϕ+ , the k
ψ
0 ⊂ g N (:= the Lie algebra of G N ) be
the linear space consisting of those generators ξ ∈ g N , for which
T ψ (P P
ϕ
+ )X ξ ψ := i
d
dt
t=0
P P
ϕ
+ exp(−it X ξ )ψ = 0.
(4.3.24)
Clearly, dim k
ψ
0 = dim G N − m ϕ is constant for all ψ ∈
O
m
ϕ+ . The equation
(4.3.24) is equivalent to the equation
(I H − P ψ (+) )P + X ξ ψ = 0, with ψ
(+)
:= P
ϕ
+ ψ.
(4.3.25)
By the relation ψ
(±)
∈ H
±
N is defined the completely symmetric (resp. antisymmetric) part of the vector ψ ∈ H N . Let
5 This vector space is, as could be seen from the formula, the image of the tangent space T ϕ
O ϕ+
by the tangent map of the mapping P P
ϕ
+ .