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1.5. Adjoint orbits in the general case
1.5 Adjoint orbits in the general case
Let us now classify Iε-adjoint orbits of gε in the general case as follows. Let C be an element
in gε and consider the Fitting decomposition of C
C m = VN ⊕VI,
where VN and VI are stable by C, CN = C|VN is nilpotent and CI = C|VI
is invertible. Since C
is skew-symmetric, Bε(C k (VN),VI) = (−1) k Bε(VN,C k (VI)) for any k then one has VI = (VN) ⊥ .
Also, the restrictions B N ε = Bε|VN×VN and BI ε = Bε|VI×VI are non-degenerate. Clearly, CN ∈
gε(VN) and CI ∈ gε(VI). By Section 1.2 and Section 1.4, CN is attached with a partition [d] ∈
Pε(n) and CI corresponds to a triple T ∈ Jℓ where n = dim(VN), 2ℓ = dim(VI). Let D(m) be
the set of all pairs ([d],T ) such that [d] ∈ Pε(n) and T ∈ Jℓ satisfying n + 2ℓ = m. By the
preceding remarks, there exists a map p : gε → D(m) . Denote by O(gε) the set of Iε-adjoint
orbits of gε then we obtain the classification of O(gε) as follows:
Proposition 1.5.1. The map p : gε → D(m) induces a bijection p : O(gε) → D(m).
Proof. Let C and C ′ be two elements in gε. Assume that C and C ′ lie in the same Iε-adjoint
orbit. It means that there exists an isometry P such that C ′ = PCP−1 . So C ′k P = P Ck for
any k in N. As a consequence, P(VN) ⊂ V ′ N and P(VI) ⊂ V ′ . However, P is an isometry then
V ′ N = P(VN) and V ′
I = P(VI). Therefore, one has
C ′ N = PN CNP −1
N and C′ I = PI CIP −1
I ,
where PN = P : VN → V ′ N and PI = P : VI → V ′
I are isometries. It implies that CN, C ′ N have the
same partition and CI, C ′ I have the same triple. Hence, the map p is well defined.
For a pair ([d],T ) ∈ D(m) with [d] ∈ Pε(n) and T ∈ Jℓ, we set a nilpotent map CN ∈
gε(VN) corresponding to [d] as in Section 1.2 and an invertible map CI ∈ gε(VI) as in Proposition
1.4.4 where dim(VN) = n and dim(VI) = 2ℓ. Define C ∈ gε by C(XN + XI) = CN(XN) +CI(XI),
for all XN ∈ VN, XI ∈ VI. By construction, p(C) = ([d],T ) and p is onto.
To prove p is one-to-one, let C,C ′ ∈ gε such that p(C) = p(C ′ ) = ([d],T ). Keep the above
notations, since CN and C ′ N have the same partition then there exists an isometry PN : VN → V ′ N
such that C ′ N = PN CN P −1
N . Similarly CI and C ′ I have the same triple and then there exists an
isometry PI : VI → V ′
I such that C′ I = PI CI P −1
I . Define P : V → V by P(XN + XI) = PN(XN) +
PI(XI), for all XN ∈ VN,XI ∈ VI then P is an isometry and C ′ = P C P−1 . Therefore, p is one-toone.
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