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

A topological Maslov index for 3-graded Lie groups 7
We call an element x ∈ V invertible if the operator
Q(x): V → V, y ↦→ Q(x)(y) := {x, y, x}
is invertible and write V × for the set of invertible elements in V . For x ∈ V ×
the (Jordan triple) inverse is defined by
The elements of the set
x ♯ := Q(x) −1 .x.
S := {x ∈ V × : x ♯ = x} = {x ∈ V × : {x, x, x} = x}
are called involutions or invertible tripotents (cf. Definition A.1).
Definition I.5. (a) We have seen above that the multiplication map G + ×
G 0 ×G − → G is a diffeomorphism onto an open subset of the group G. Therefore
we have smooth maps
pj: G + G 0 G − → G j with g = p+(g)p0(g)p−(g) for g ∈ G + G 0 G − .
For z ∈ g1 and g ∈ G with g exp z ∈ G + G 0 G − we define
JG(g, z) := p0(g exp z) ∈ G 0 .
The function JG is called the universal automorphy factor of G.
(b) For g ∈ G we put g∗ := τ(g) −1 and for x ∈ g we put x∗ := −τ.x. For
w ∈ g1 and g = (exp w) ∗ = exp w∗ ∈ G− we then set
∗ −1 ∗ −1
BG(z, w) := JG (expw) , z) = p0 (exp w) expz ∈ G0
whenever expw ∗ exp z ∈ G + G 0 G − . According to Lemma I.3, this happens if
and only if the Bergman operators
B(v, w) := B+(v, w ∗ ) = idV +adv ad w ∗ + 1
4 (ad v)2 (ad w ∗ ) 2
= idV +adv ad w ∗ + 1
4 (adv)2 ◦ τ ◦ (adw) 2 ◦ τ = idV −2vw + Q(v)Q(w)
and B(w, v) are invertible. In this case the pair (v, w) ∈ V 2 is called quasiinvertible
and we write v⊤w to denote quasi-invertibility. This notation is
motivated by the fact that, in terms of Appendix B, quasi-invertibility of (v, w)
is equivalent to (exp(−τ.w) expv.f−)⊤f+, which means that the 3-filtration
exp v.f− is transversal to the 3-filtration exp(τ.w).f+ = τX(expw.f−).
(c) We write
V 2 ⊤ := {(x, y) ∈ V 2 : B(x, y), B(y, x) ∈ GL(V )}
for the set of quasi-invertible pairs in V , and V 3 ⊤ := {(x, y, z) ∈ V 3 : (x, y), (y, z), (x, z) ∈
V 2 ⊤ } for the set of quasi-invertible triples. For the set S of involutions in V we
. We then consider the functions
put S2 ⊤ := S2 ∩ V 2 ⊤ and S3 ⊤ := S3 ∩ V 3 ⊤
cG: V 3 ⊤ → G 0 , cG(x, y, z) := BG(x, y)BG(z, y) −1 BG(z, x)
and dG: V 3 ⊤ → G0 , (x, y, z) ↦→ cG(x, y, z)cG(x, z, y) −1 with
dG(x, y, z) = BG(x, y)BG(z, y) −1 BG(z, x)BG(y, x) −1 BG(y, z)BG(x, z) −1 .
Lemma I.6. For a quasi-invertible pair (v, w) in V and the adjoint representation
ρV : G 0 → GL(V ) of G 0 on g1 = V we have B(v, w) = ρV (BG(v, w)).
Proof. This follows from the proof of Theorem 2.10 in [BN04a].