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

1.2. Nilpotent orbits
• For p ≥ 1 we equip the vector space C2p+1 with its canonical bilinear form B1 and the
map CJ 2p+1 having the matrix
C J 2p+1 =
Jp+1 M
0 −t
Jp
in a canonical basis where M = (mi j) denotes the (p + 1) × p-matrix with mp+1,p = −1
and mi j = 0 otherwise. Then CJ 2p+1 ∈ o(2p + 1)
• For p ≥ 1, we consider the vector space C 2p equipped with its canonical bilinear form
B−1 and the map C J p+p with matrix
Jp M
0 − t Jp
in a canonical basis where M = (mi j) denotes the p× p-matrix with mp,p = 1 and mi j = 0
otherwise. Then C J p+p ∈ sp(2p).
For each partition [d] ∈ P−1(2n), [d] can be written as
(p1, p1, p2, p2,..., pk, pk,2q1,...,2qℓ)
with all pi odd, p1 ≥ p2 ≥ ··· ≥ pk and q1 ≥ q2 ≥ ··· ≥ qℓ. We associate a map C [d] with the
matrix:
diag k+ℓ (C J 2p1 ,CJ 2p2 ,...,CJ 2pk ,CJ q1+q1 ,...,CJ qℓ+qℓ )
in a canonical basis of C 2n then C [d] ∈ sp(2n).
Similarly, let [d] ∈ P1(m), [d] can be written as
(p1, p1, p2, p2,..., pk, pk,2q1 + 1,...,2qℓ + 1)
with all pi even, p1 ≥ p2 ≥ ··· ≥ pk and q1 ≥ q2 ≥ ··· ≥ qℓ. We associate a map C [d] with the
matrix:
diag k+ℓ (C J 2p1 ,CJ 2p2 ,...,CJ 2pk ,CJ 2q1+1 ,...,CJ 2qℓ+1 ).
in a canonical basis of C m then C [d] ∈ o(m).
By Proposition 1.2.10, it is sure that our construction is a bijection between the set Pε(m)
and the set of nilpotent Iε-adjoint orbits in gε.
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