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1.2 Nilpotent orbits
1.2. Nilpotent orbits
Let n ∈ N ∗ , a partition [d] of n is a tuple [d1,...,dk] of positive integers satisfying
d1 ≥ ... ≥ dk and d1 + ... + dk = n.
Occasionally, we use the notation [t i1
1 ,...,tir r ] to replace the partition [d1,...,dk] where
⎧
⎪⎨
t1 1 ≤ j ≤ i1
t2 i1 + 1 ≤ j ≤ i1 + i2
d j =
⎪⎩
t3 i1 + i2 + 1 ≤ j ≤ i1 + i2 + i3
...
Each i j is called the multiplicity of t j. Denote by P(n) the set of partitions of n. For example,
P(3) = {[3], [2,1], [13 ]} and P(4) = {[4], [3,1], [22 ], [2,12 ], [14 ]}.
Let p ∈ N∗ . We denote the Jordan block of size p by J1 = (0) and for p ≥ 2,
⎛
0
⎜
⎜0
⎜
Jp := ⎜
.
⎝0
1
0
.
0
0
1
...
...
...
...
. ..
0
⎞
0
0 ⎟
. ⎟.
⎟
1⎠
0 0 0 ... 0
Then Jp is a nilpotent endomorphism of C p . Given a partition [d] = [d1,...,dk] ∈ P(n) there is
a nilpotent endomorphism of C n defined by
X [d] := diagk (Jd1 ,...,Jdk ).
Moreover, X [d] is also a nilpotent element of sl(n) since its trace is zero. Conversely, if C is a
nilpotent element in sl(n) then C is GL(n)-conjugate to its Jordan normal form X [d] for some
partition [d] ∈ P(n).
Given two different partitions [d] = [d1,...,dk] and [d ′ ] = [d ′ 1 ,...,d′ l ] of n then the GL(n)adjoint
orbits through X [d] and X [d ′ ] respectively are disjoint by the unicity of Jordan normal
form. Therefore, one has the following proposition:
Proposition 1.2.1. There is a one-to-one correspondence between the set of nilpotent GL(n)adjoint
orbits of sl(n) and the set P(n).
It results that sl(n) has only finitely many nilpotent GL(n)-adjoint orbits, exactly ♯P(n).
However, this does not assure the same for its semisimple subalgebras and the classification of
nilpotent adjoint orbits of gε is rather more difficult since the action of the subgroup Iε does not
coincide with the action of GL(n). However, by many works of Dynkin, Kostant and Mal’cev
(see [CM93]), there is an important bijection between nilpotent adjoint orbits of a semisimple
Lie algebra g and a subset of 3 rank(g) possible weight Dynkin diagrams where rank(g) is the
dimension of a Cartan subalgebra of g, and thus gε has only finitely many nilpotent adjoint
orbits.
The main tool in the classical work on nilpotent adjoint orbits is the representation theory
of the Lie algebra sl(2) (or sl(2)-theory, for short) applied to the adjoint action on a semisimple
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