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The symplectic group, that is the group of linear automorphisms of T ∗Ên preserving ω0, is
denoted by Sp(2n). Let L (n) be the Grassmannian of Lagrangian subspaces of T ∗Ên , that is the
set of n-dimensional linear subspaces of T ∗Ên on which ω0 vanishes. The relative Maslov index
assigns to every pair of Lagrangian paths λ1, λ2 : [a, b] → L (n) a half integer µ(λ1, λ2). We refer
to [RS93] for the definition and for the properties of the relative Maslov index.
Another useful invariant is the Hörmander index of four Lagrangian subspaces (see [Hör71],
[Dui76], or [RS93]):
5.1. Definition. Let λ0, λ1, ν0, ν1 be four Lagrangian subspaces of T ∗Ên . Their Hörmander index
is the half integer
h(λ0, λ1; ν0, ν1) := µ(ν, λ1) − µ(ν, λ0),
where ν : [0, 1] → L (n) is a Lagrangian path such that ν(0) = ν0 and ν(1) = ν1.
Indeed, the quantity defined above does not depend on the choice of the Lagrangian path ν
joining ν0 and ν1.
If V is a linear subspace ofÊn , N ∗ V ⊂ T ∗Ên denotes its conormal space, that is
N ∗ V := {(q, p) ∈Ên × (Ên ) ∗ | q ∈ V, V ⊂ kerp} = V × V ⊥ ,
where V ⊥ denotes the set of covectors in (Ên ) ∗ which vanish on V . Conormal spaces are Lagrangian
subspaces of T ∗Ên .
Let C : T ∗Ên → T ∗Ên be the linear involution
C(q, p) := (q, −p) ∀(q, p) ∈ T ∗Ên .
The involution C is anti-symplectic, meaning that
ω0(Cξ, Cη) = −ω0(ξ, η) ∀ξ, η ∈ T ∗Ên .
In particular, C maps Lagrangian subspaces into Lagrangian subspaces. Since the Maslov index
is natural with respect to symplectic transformations and changes sign if we change the sign of
the symplectic structure, we have the identity
µ(Cλ, Cν) = −µ(λ, ν), (76)
for every pair of Lagrangian paths λ, ν : [a, b] → L (n). Since conormal subspaces are C-invariant,
we deduce that
µ(N ∗ V, N ∗ W) = 0, (77)
for every pair of paths V, W into the Grassmannian ofÊn . Let V0, V1, W0, W1 be four linear
subspaces ofÊn , and let ν : [0, 1] → L (n) be a Lagrangian path such that ν(0) = N ∗ W0 and
ν(1) = N ∗ W1. By (76),
h(N ∗ V0, N ∗ V1; N ∗ W0, N ∗ W1) = µ(ν, N ∗ V1) − µ(ν, N ∗ V0) = −µ(Cν, N ∗ V1) + µ(Cν, N ∗ V0).
But also the Lagrangian path Cν joins N ∗ W0 and N ∗ W1, so the latter quantity equals
We deduce the following:
−h(N ∗ V0, N ∗ V1; N ∗ W0, N ∗ W1).
5.2. Proposition. Let V0, V1, W0, W1 be four linear subspaces ofÊn . Then
h(N ∗ V0, N ∗ V1; N ∗ W0, N ∗ W1) = 0.
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