A topological Maslov index for 3-graded Lie groups - ResearchGate

A topological Maslov index for 3-graded Lie groups 5
(3) X := G/P − is a homogeneous Banach manifold and the map g1 → X, x ↦→
exp xP − is a diffeomorphism onto an open subset.
(4) The orbits of the identity component G0 of G coincide with the connected
components of X.
(5) For the inner 3-filtrations f± = (g±, g± + g0) of g we have Gf± = P ± and
hence an embedding
(1.1) X → F, gP − ↦→ g.f−
of X into the set F of inner 3-filtrations of g.
Proof. (1) Since G 0 preserves the grading of g, it normalizes the subgroups
G ± , so that P ± are groups.
We consider the two inner 3-filtrations
f+ := (g+, g+ + g0) and f− := (g−, g− + g0)
defined by the 3-grading of g (cf. Appendix B for the definitions concerning
inner 3-filtrations). For a 3-filtration f = (f1, f0) let
Gf := {g ∈ G: Ad(g).f0 = f0, Ad(g).f1 = f1}
denote its stabilizer subgroup in G. Then we clearly have P ± ⊆ Gf± .
On the other hand each element g ∈ Gf+ also stabilizes the subset f⊤ + =
{e ∈ F: e⊤f+} of all inner 3-filtrations of g transversal to f+. According to
[BN04a, Th. 1.6(2)], the group G + acts transitively on the set f⊤ + containing f−.
Hence there exists an element g+ ∈ G + with g.f− = g+.f−. Then g −1
+ g.f± = f±
implies that g −1
+ g also preserves the 3-grading given by
g+ = f+,1, g− = f−,1 and g0 = f+,0 ∩ f−,0.
Therefore g −1
+ g ∈ G0 , so that g ∈ g+G0 ⊆ P + . This shows that P + = Gf+ and
likewise we get P − = Gf− . From that we obtain
P + ∩ P − = Gf+ ∩ Gf− = G0 .
Let E ∈ g0 be a grading element, i.e., gj is the j-eigenspace of ad E.
Then we have for x ∈ g+ the relation
Ad(expx).E = e ad x .E = E − [x, E] = E + x.
Since this element is contained in g− + g0 = f−,0 if and only if x = 0, we get
G + ∩ P − = G + ∩ Gf−
= {1},
and likewise G − ∩ P + = {1}.
From P ± = Gf± we derive in particular that P ± and G 0 are Lie subgroups
of g with the Lie algebras p ± = g++g0 which are the normalizers of the flags f±