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

1.2. Nilpotent orbits
The point of the above proof is that the Killing form of g is non-degenerate and invariant.
However, the existence of a sl(2)-triple {H,X,Y } having the nilpositive element X is not unique,
that is, it may exist another sl(2)-triple {H ′ ,X,Y ′ } with the same nilpositive element. The
following theorem shows that our choice is is unique up to an element in Gad, the identity
component of the automorphism group Aut(g) = {φ ∈ GL(g) | [φ(X),φ(Y )] = φ([X,Y ])}.
Proposition 1.2.8 (Kostant).
Let g be a complex semisimple Lie algebra. Any two sl(2)-triples {H,X,Y } and {H ′ ,X,Y ′ }
with the same nilpositive element are conjugate by an element of Gad.
Denote by Atriple the set of sl(2)-triples of g, Atriple the set of Gad-conjugacy classes of
sl(2)-triples in Atriple and N (g) the set of non-zero nilpotent orbits in g then we obtain the
corollary:
Corollary 1.2.9. The map ω : Atriple → N (g) defined by ω({H,X,Y }) = OX induces a bijection
Ω : Atriple → N (g).
Proof. Let O be a non-zero nilpotent orbit in g. Fix X = 0 in O. By Proposition 1.2.6, there
exists a sl(2)-triple {H,X,Y } such that X is its nilpositive element so Ω is onto. If there exists
another sl(2)-triple {H ′ ,X,Y ′ } such that X is also nilpositive then by Proposition 1.2.8
{H ′ ,X,Y ′ } must lie in Gad-conjugacy class of {H,X,Y }. Therefore, Ω is one-to-one.
Now we turn to our problem of classification nilpotent Iε-adjoint orbits of gε. Define the set
Pε(m) = {[d1,...,dm] ∈ P(m)| ♯{ j | d j = i} is even for all i such that (−1) i = ε}.
In particular, P1(m) is the set of partitions of m in which even parts occur with even multiplicity
and P−1(m) is the set of partitions of m in which odd parts occur with even multiplicity.
Proposition 1.2.10 (Gerstenhaber).
Nilpotent Iε-adjoint orbits in gε are in one-to-one correspondence with the set of partitions
in Pε(m).
Proof. A proof of the proposition can be found in [CM93], Theorem 5.1.6.
Here, we give the construction of a nilpotent element in gε from a partition [d] of m that is
useful for next two chapters. Define maps in gε as follows:
• For p ≥ 2, we equip the vector space C2p with its canonical bilinear form Bε and the map
CJ 2p having the matrix
C J 2p =
Jp
0
0
−t
Jp
in a canonical basis where t Jp denotes the transpose matrix of the Jordan block Jp. Then
C J 2p ∈ gε(C 2p ).
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