TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...

1.4. Invertible orbits
Proof. Let C and C ′ ∈ I (2n) such that C ′ = U C U −1 with U ∈ Iε. Let S, S ′ , N, N ′ be respectively
the semisimple and nilpotent parts of C and C ′ . Write i(C) = (Λ,m,d) and i(C ′ ) =
(Λ ′ ,m ′ ,d ′ ). One has
S ′ + N ′ = U (S + N) U −1 = U S U −1 +U N U −1 .
By the unicity of Jordan decomposition, S ′ = U S U −1 and N ′ = U N U −1 . So Λ ′ = Λ and
m ′ = m. Also, since U S = S ′ U one has U S(V λ ) = S ′ U(V λ ). It implies that
S ′ (U(V λ )) = λU(V λ ).
That means U(V λ ) = V ′ λ , for all λ ∈ Λ. Since N′ = U N U −1 then N|V λ and N ′ | V ′ λ have the same
Jordan decomposition, so d = d ′ and i is well defined.
To prove that i is onto, we start with Λ = {λ1,−λ1,...,λk,−λk}, m and d as in Definition
1.4.3. Define on the canonical basis:
m(λ1)
m(λk)
m(λ1)
m(λk)
S = diag2n ( λ1,...,λ1,..., λk,...,λk, −λ1,...,−λ1,..., −λk,...,−λk).
For all 1 ≤ i ≤ k, let d(λi) = (d1(λi) ≥ ··· ≥ dr (λi) ≥ 1) and define
λi
N+(λi) = diagd(λi) Jd1(λi),J d2(λi),...,J dr (λi)
λi
on the eigenspace Vλi and 0 on the eigenspace V−λi where Jd is the Jordan block of size d.
By Lemma 1.4.2, N(λi) = N+(λi) − N∗ +(λi) is skew-symmetric on Vλi ⊕V−λi . Finally,
C 2n =
⊥
1≤i≤k
Vλi ⊕V
−λi .
Define N ∈ gε by N ∑ k i=1 vi
k = ∑i=1 N(λi)(vi), vi ∈ Vλi ⊕V−λi and C = S + N ∈ gε. By construction,
i(C) = (Λ,m,d), so i is onto.
To prove that i is one-to-one, assume that C, C ′ ∈ I (2n) and that i(C) = i(C ′ ) = (Λ,m,d).
Using the previous notation, since their respective semisimple parts S and S ′ have the same
spectrum and same multiplicities, there exist U ∈ Iε such that S ′ = USU −1 . For λ ∈ Λ, we have
U(Vλ ) = V ′ λ for eigenspaces Vλ and V ′ λ of S and S′ respectively.
Also, for λ ∈ Λ, if N and N ′ are the nilpotent parts of C and C ′ , then N ′′ (Vλ ) ⊂ Vλ , with
N ′′ = U−1N ′ U. Since i(C) = i(C ′ ), then N|V and N λ ′ | V ′ have the same Jordan type. Since
λ
N ′′ = U −1 N ′ U, then N ′′ |V λ and N ′ | V ′ λ have the same Jordan type. So N|V λ and N ′′ |V λ have the
same Jordan type. Therefore, there exists D+ ∈ L (Vλ ) such that N ′′ |V = D+N|V D λ λ −1
+ . By
Lemma 1.4.2, there exists D(λ) ∈ Iε(Vλ ⊕V−λ ) such that
N ′′ |V λ ⊕V −λ = D(λ)N|V λ ⊕V −λ D(λ) −1 .
We define D ∈ Iε by D|V λ ⊕V −λ = D(λ), for all λ ∈ Λ. Then N ′′ = DND −1 and D commutes with
S since S|V ±λ is scalar. Then S ′ = (UD)S(UD) −1 and N ′ = (UD)N(UD) −1 and we conclude
C ′ = (UD)C(UD) −1 .
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