§3. Coadjoint invariant functions 15

3.2. Examples.

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

Rn

there exists a cyclic vector ξ, i.e. such that the vectors ξ, Aξ,

A2ξ,...,

An−1ξ

form a basis in

Rn.

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 ·

n+1.−

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

on X.

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

g∗

form the center of the Lie algebra

C∞(g∗)

with respect to Poisson brackets.