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

1.1. Definitions
and gε(V ) = {C ∈ L (V ) | C is skew-symmetric}.
Then Iε(V ) is the isometry group of the bilinear form Bε and gε(V ) is its Lie algebra. Denote
by A ∗ ∈ L (V ) the adjoint map of an element A ∈ L (V ) with respect to Bε, then A ∈ Iε(V ) if
and only if A −1 = A ∗ and C ∈ gε(V ) if and only if C ∗ = −C. If ε = 1 then Iε(V ) is denoted by
O(V ) and gε(V ) is denoted by o(V ). If ε = −1 then Sp(V ) stands for Iε(V ) and sp(V ) stands
for gε(V ).
Recall that the adjoint action Ad of Iε(V ) on gε(V ) is given by
AdU(C) = UCU −1 , ∀ U ∈ Iε(V ), C ∈ gε(V ).
We denote by OC = Ad Iε(V )(C), the adjoint orbit of an element C ∈ gε(V ) by this action.
If V = C n , we call Bε a canonical bilinear form of C n . And with respect to Bε, we define
a canonical basis B = {E1,...,Em} of C m as follows. If m even, m = 2n, write B =
{E1,...,En,F1,...,Fn}, if m is odd, m = 2n + 1, write B = {E1,...,En,G,F1,...,Fn} and one
has:
• if m = 2n then
where 1 ≤ i, j ≤ n.
B1(Ei,Fj) = B1(Fj,Ei) = δi j, B1(Ei,Ej) = B1(Fi,Fj) = 0,
B−1(Ei,Fj) = −B−1(Fj,Ei) = δi j, B−1(Ei,Ej) = B−1(Fi,Fj) = 0,
• if m = 2n + 1 then ε = 1 and
⎧
⎪⎨ B1(Ei,Fj) = δi j, B1(Ei,Ej) = B1(Fi,Fj) = 0,
B1(Ei,G) = B1(Fj,G) = 0,
⎪⎩
B1(G,G) = 1
where 1 ≤ i, j ≤ n.
Also, in the case V = C m , we denote by GL(m) instead of GL(V ), O(m) stands for O(V )
and o(m) stands for o(V ). If m = 2n then Sp(2n) stands for Sp(V ) and sp(2n) stands for sp(V ).
We will also write Iε = Iε(C m ) and gε = gε(C m ). The goal of this chapter is classifying all of
Iε-adjoint orbits of gε.
Finally, let V is an m-dimensional vector space. If V is quadratic then V is isometrically
isomorphic to the quadratic space (C m ,B1) and if V is symplectic then V is isometrically isomorphic
to the symplectic space (C m ,B−1) [Bou59].
4