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We identify the product T ∗Ên ×T ∗Ên with T ∗Ê2n , and we endow it with its standard symplectic
structure. In other words, we consider the product symplectic form, not the twisted one used in
[RS93]. Note that the conormal space of the diagonal ∆Ên inÊn ×Ên is the graph of C,
N ∗ ∆Ên = graphC ⊂ T ∗Ên × T ∗Ên = T ∗Ê2n .
The linear endomorphism Ψ of T ∗Ên belongs to the symplectic group Sp(2n) if and only if the
graph of the linear endomorphism ΨC is a Lagrangian subspace of T ∗Ê2n , if and only if the graph
of CΨ is a Lagrangian subspace of T ∗Ê2n . If λ1, λ2 are paths of Lagrangian subspaces of T ∗Ên
and Ψ is a path in Sp(2n), Theorem 3.2 of [RS93] leads to the identities
µ(Ψλ1, λ2) = µ(graph(ΨC), Cλ1 × λ2) = −µ(graph(CΨ), λ1 × Cλ2). (78)
The Conley-Zehnder index µCZ(Ψ) of a symplectic path Ψ : [0, 1] → Sp(2n) is related to the
relative Maslov index by the formula
µCZ(Ψ) = µ(N ∗ ∆Ên, graphCΨ) = µ(graphΨC, N ∗ ∆Ên). (79)
We conclude this section by fixing some standard identifications, which allow to see T ∗Ên as
∗Ên a complex vector space. By using the Euclidean inner product onÊn , we can identify T with
Ê2n . We also identify the latter space to�n , by means of the isomorphism (q, p) ↦→ q + ip. In
other words, we consider the complex structure
� �
0 −I
J0 :=
I 0
onÊ2n . With these identifications, the Euclidean inner product u · v, respectively the symplectic
product ω0(u, v), of two vectors u, v ∈ T ∗Ên ∼ =Ê2n ∼ =�n is the real part, respectively the
imaginary part, of their Hermitian product 〈·, ·〉,
〈u, v〉 :=
n�
ujvj = u · v + i ω0(u, v).
j=1
The involution C is the complex conjugacy. By identifying V ⊥ with the Euclidean orthogonal
complement, we have
If λ : [0, 1] → L (n) is the path
N ∗ V = V ⊕ iV ⊥ = � z ∈�n | Re z ∈ V, Im z ∈ V ⊥ � .
λ(t) = e iαtÊ, α ∈Ê,
the relative Maslov index of λ with respect toÊis the half integer
�
1 α −
µ(λ,Ê) = 2 − ⌊ π ⌋ if α ∈Ê\π�,
if α ∈ π�.
− α
π
Notice that the sign is different from the one appearing in [RS93] (localization axiom in Theorem
2.3), due to the fact that we are using the opposite symplectic form onÊ2n . Our sign convention
here also differs from the one used in [AS06b], because we are using the opposite complex structure
onÊ2n .
5.2 Elliptic estimates on the quadrant
We recall that a real linear subspace V of�n is said totally real if V ∩iV = (0). Denote byÀthe upper
half-plane {z ∈�|Im z > 0}, and byÀ+ the upper-right quadrant {z ∈�|Re z > 0, Im z > 0}.
We shall make use of the following Calderon-Zygmund estimates for the Cauchy-Riemann operator
∂ = ∂s + i∂t:
56
(80)