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

A topological Maslov index for 3-graded Lie groups 21
so that v ∈ V−. If, conversely, v ∈ V × − , then e − v is invertible (Lemma III.1)
and z := Ce(v) ∈ S is a unitary element for which z + e is invertible. Since v
is invertible, Lemma A.11 implies that z = Ce(v) lies in the domain V × + e of
Ce, so that also e − z is invertible, and hence (e, −e, z) ∈ S 3 ⊤ .
Remark III.3. (a) The preceding lemma shows in particular that (e, f) ∈ S 2 ⊤
implies that f ∈ S−e = −Se, where Se denotes the connected component of S
containing e. For (e, f, g) ∈ S 3 ⊤ we even conclude that Se = S−g = Sf = S−e =
Sg. This leads to the disjoint decomposition
S 3 ⊤ =
(Se) 3 ⊤,
e
so that it is no loss of generality if we consider only a fixed connected component
Se of the set S and study the index map on the subset (Se) 3 ⊤ of S3 ⊤ .
we have
(b) For G = PSL2(R) = SL2(R)/{±1} with grading derivation
−1 1 0
a b a −c
D = ad and τ =
0 −1
c d −b d
V ∼ = R with {x, y, z} = xyz and S = {±1}.
Here S2 ⊤
case we have S1 = {1} = {−1} = S−1.
= {(1, −1), (−1, 1)} and the connected group H acts trivially. In this
Lemma III.4. Fix e ∈ S and consider the associated Cayley transform C :=
Ce ∈ Aut(g). Then the involution τC := CτC −1 ∈ Aut(g) satisfies:
(1) τC preserves the 3-grading of g.
(2) τC = τC2 , where τ and C2 are commuting involutions of g.
(3) The Lie subalgebra l := C(gτ ) = gτC is adapted to the 3-grading of g and
τ -invariant.
(4) τC |V = −Q(e).
(5) For the stabilizer group He,−e, the identity component L := (GτC CG
π = exp (e − f) ∈ G we have
4
Ad(C G ) = C and L 0 := L ∩ G 0 = C G · He,−e · (C G ) −1 .
Proof. (1) With Lemma I.11 we get in X ⊆ F :
)0 and
τ C (f−) = τ C (0) = CτX(−e) = C(−e) = 0 = f− and τ C (f+) = CτX(e) = C(e) = f+.
Therefore τ C fixes the two filtrations f± and hence the corresponding 3-grading
of g.
(2) With Lemma I.11 we get τ C = CτC −1 = C 2 τ = τC −2 = τC 2 , so that
the two involutions τ and C 2 commute.
(3) That l is adapted to the 3-grading of g follows directly from (1). Since
τ and τ C commutes by (2), l is τ -invariant.
(4) follows from Lemma I.11(4).
(5) The relation Ad(C G ) = C is immediate from the definitions. Further
C(±e) = f± and C G H(C G ) −1 = L lead to L 0 = L∩Gf± = CG ·He,−e ·(C G ) −1 .