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

1.4. Invertible orbits
Let us now consider C ∈ gε, C invertible. Then, m must be even (obviously, it happened if
ε = −1), m = 2n (see Appendix A). We decompose C = S + N into semisimple and nilpotent
parts, S, N ∈ gε by its Jordan decomposition. It is clear that S is invertible. We have λ ∈ Λ if
and only if −λ ∈ Λ (Appendix A) where Λ is the spectrum of S. Also, m(λ) = m(−λ), for all
λ ∈ Λ with the multiplicity m(λ). Since N and S commute, we have N(V ±λ ) ⊂ V ±λ where V λ
is the eigenspace of S corresponding to λ ∈ Λ. Denote by W(λ) the direct sum
Then
Define the equivalence relation R on Λ by:
W(λ) = V λ ⊕V −λ .
λRµ if and only if λ = ±µ.
C 2n =
⊥
λ∈Λ/R
W(λ),
and each (W(λ),B λ ) is a vector space with the non-degenerate form B λ = Bε| W(λ)×W(λ).
Fix λ ∈ Λ. We write W(λ) = V+ ⊕V− with V± = V ±λ . Then, according to the notation in
Lemma 1.4.2, define N ±λ = N±. Since N|V− = −N∗ λ , it is easy to verify that the matrices of
N|V+ and N|V− have the same Jordan form. Let (d1(λ),...,dr (λ)) be the size of the Jordan
λ
blocks in the Jordan decomposition of N|V+ . This does not depend on a possible choice between
N|V+ or N|V− since both maps have the same Jordan type.
Next, we consider
D =
{(d1,...,dr) ∈ N r | d1 ≥ d2 ≥ ··· ≥ dr ≥ 1}.
r∈N ∗
Define d : Λ → D by d(λ) = (d1(λ),...,dr λ (λ)). It is clear that Φ ◦ d = m where Φ : D → N
is the map defined by Φ(d1,...,dr) = ∑ r i=1 di.
Finally, we can associate to C ∈ gε a triple (Λ,m,d) defined as above.
Definition 1.4.3. Let Jn be the set of all triples (Λ,m,d) such that:
(1) Λ is a subset of C \ {0} with ♯Λ ≤ 2n and λ ∈ Λ if and only if −λ ∈ Λ.
(2) m : Λ → N∗ satisfies m(λ) = m(−λ), for all λ ∈ Λ and ∑ m(λ) = 2n.
λ∈Λ
(3) d : Λ → D satisfies d(λ) = d(−λ), for all λ ∈ Λ and Φ ◦ d = m.
Let I (2n) be the set of invertible elements in gε and I (2n) be the set of Iε-adjoint orbits
of elements in I (2n). By the preceding remarks, there is a map i : I (2n) → Jn. Then we
have a parametrization of the set I (2n) as follows:
Proposition 1.4.4.
The map i : I (2n) → Jn induces a bijection i :
I (2n) → Jn.
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