§3. Coadjoint invariant functions 15
Example 3. Let G = GL(n, R) act on X = Matn(R) by conjugation.
Let S be the aﬃne subset consisting of matrices of the form
0 1 0 · · · 0
0 0 1 · · · 0
· · · · · · ·
0 0 0 · · · 1
cn cn−1 cn−2 · · · c1
One can check that S intersects almost all conjugacy classes in exactly
one point. Geometrically this means that for almost all operators A on
there exists a cyclic vector ξ, i.e. such that the vectors ξ, Aξ,
form a basis in
It is clear that in this basis the matrix of A has
the above form.
Using the section S we show that in this case polynomial invariants form
an algebra R[c1, c2,...,cn]. Indeed, every polynomial invariant restricted
to S becomes a polynomial in c1,c2,...,cn. On the other hand, all the
ci’s admit extensions as invariant polynomials on Matn(R). Namely, they
coincide up to sign with the coeﬃcients of the characteristic polynomial
PA(λ) = det(A − λ · 1).
There is a nice generalization of this example to all semisimple Lie alge-
bras due to Kostant (see [Ko2]). ♦
Example 4. Let N+ (resp. N−) be the subgroup of strictly upper (resp.
lower) triangular matrices from GL(n, R). The group G = N+ × N− acts on
X = Matn(R):
g = (n+, n−) : A → n− · A ·
Take the subspace of diagonal matrices as S. Then almost all G-orbits
intersect S in a single point (the Gauss Lemma in linear algebra). But in
this case polynomial functions on S extend to rational invariant functions
Namely, let Δk(A) denote the principal minor of order k for a matrix A.
It is a G-invariant polynomial on X. Denote by fk the function on S that
is equal to the k-th diagonal element. Then the restriction of Δk to S is the
product f1f2 · · · fk. We see that the function fk extends to X as a rational
function Δk/Δk−1. ♦
In the case of the coadjoint action the polynomial and rational invariants
play an important role in representation theory due to their connection with
infinitesimal characters (see the next chapters). Here we remark only that
smooth K(G)-invariants on
form the center of the Lie algebra
with respect to Poisson brackets.