§1. Basic definitions 3
Matn(R) onto V parallel to
g⊥.
Then the coadjoint representation K can
be written in a simple form
(4) K(g): F pV
(gFg−1).
Remark 1. If we could choose V invariant under Ad (G) (which we
can always assume for g semisimple or reductive), then we can omit the
projection pV in (4).
Example 1. Denote by G the group of all (non-strictly) upper trian-
gular matrices g GL(n, R), i.e. such that gij = 0 for i j. Then the
Lie algebra g consists of all upper triangular matrices from Matn(R). The
space
g⊥
is the space of strictly upper triangular matrices X satisfying the
condition xij = 0 for i j.
We can take for V the space of all lower triangular matrices.
The projection pV in this case sends any matrix to its “lower part” (i.e.
replaces all entries above the main diagonal by zeros). Hence, the coadjoint
representation takes the form
K(g) : F (g
Fg−1)lower
part.
Although this example has been known for a long time and has been
thoroughly studied by many authors, we still do not know how to classify
the coadjoint orbits for general n.
Example 2. Let G = SO(n, R). Then g consists of all skew-symmetric
matrices X =
−Xt
from Matn(R).
Here we can put V = g and omit the projection pV in (4) (cf. Remark
1):
K(g)X = g · X ·
g−1.
Thus, the coadjoint representation is equivalent to the adjoint one and co-
incides with the standard action of the orthogonal group on the space of
antisymmetric bilinear forms. It is well known that a coadjoint orbit passing
through X is determined by the spectrum of X, which can be any multiset
in iR, symmetric with respect to the complex conjugation. Another conve-
nient set of parameters labelling the orbits is the collection of real numbers
{tr
X2,
tr
X4,
. . . , tr
X2k}
where k = [
n
2
].
We also give the formula for the infinitesimal version of the coadjoint
action, i.e. for the corresponding representation K∗ of the Lie algebra g in
g∗:
(5) K∗(X)F, Y = F, −ad(X) Y = F, [Y, X] .
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