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

1.3 Semisimple orbits
We recall the well-known result [CM93]:
1.3. Semisimple orbits
Proposition 1.3.1. Let g be a semisimple Lie algebra, h be a Cartan subalgebra of g and W be
the associated Weyl group. Then there is a bijection between the set of semisimple orbits of g
and h/W.
For each gε, we choose the Cartan subalgebra h given by the vector space of diagonal
matrices of type
diag 2n (λ1,...,λn,−λ1,...,−λn)
if gε = o(2n) or gε = sp(2n) and of type
diag 2n+1 (λ1,...,λn,0,−λ1,...,−λn)
if gε = o(2n + 1).
Any diagonalizable (equivalently semisimple) C ∈ gε is conjugate to an element of h (see
Appendix A for a direct proof).
If gε = sp(2n) then any two eigenvectors v,w ∈ C 2n of X ∈ gε with eigenvalues λ,λ ′ ∈ C
such that λ + λ ′ = 0 are orthorgonal. Moreover, each eigenvalue pair λ,−λ is corresponding
to an eigenvector pair (v,w) satisfying Bε(v,w) = 1 and we can easily arrange for vectors v,v ′
lying in a distinct pair (v,w),(v ′ ,w ′ ) to be orthogonal, regardless of the eigenvalues involved.
That means the associated Weyl group is of all coordinate permutations and sign changes of
(λ1,...,λn). We denote it by Gn.
If gε = o(2n), the associated Weyl group, when considered in the action of the group
SO(2n), consists all coordinate permutations and even sign changes of (λ1,...,λn). However,
we only focus on O(2n)-adjoint orbits of o(2n) obtained by the action of the full orthogonal
group, then similarly to preceding analysis any sign change effects. The corresponding group
is still Gn. If gε = o(2n + 1), the Weyl group is Gn and there is nothing to add.
Now, let Λn = {(λ1,...,λn) | λ1,...,λn ∈ C, λi = 0 for some i}.
Corollary 1.3.2. There is a bijection between non-zero semisimple Iε-adjoint orbits of gε and
Λn/Gn.
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