14 1. Geometry of Coadjoint Orbits

3. Coadjoint invariant functions

3.1. General properties of invariants.

The ideology of the orbit method attaches significant importance to the

classification problem for coadjoint orbits. The first step in the solution of

this problem is to find all invariants of the group action. Polynomial and

rational invariants are especially interesting.

We also need the notion of relative invariants defined as follows. Let

X be a left G-set, and let λ be a multiplicative character of G. We say that

a function f is a relative invariant of type λ if

f(g · x) = λ(g)f(x) for all g ∈ G.

It is known that for algebraic actions of complex algebraic groups on

aﬃne algebraic manifolds there are enough rational invariants to separate

the orbits in the following sense.

Proposition 3 (see [Bor, R]). The common level set of all rational in-

variants consists of a finite number of orbits, and a generic level is just one

orbit.

Moreover, each rational invariant can be written in the form R =

P

Q

where P and Q are relative polynomial invariants of the same type.

For real algebraic groups the common level sets of invariants can split

into a finite number of connected components that are not separated by

rational invariants. (Compare with the geometry of quadrics in a real aﬃne

plane. The two branches of a hyperbola is the most visual example.)

A useful scheme for the construction of invariants of a group G acting on

a space X is the following. Suppose we can construct a subset S ⊂ X which

intersects all (or almost all) orbits in a single point. Any invariant function

on X defines, by restriction, a function on S. Conversely, any function on

S can be canonically extended to a G-invariant function defined on X (or

almost everywhere on X).

If S is smooth (resp. algebraic, rational, etc.), then we get information

about smooth (resp. algebraic, rational, etc.) invariants.

Warning. While the restriction to S usually preserves the nice prop-

erties of invariants, the extending procedure does not. For example, the

extension of a polynomial function could only be rational (see Example 5

below). ♠