10 MICHAEL FIELD

Lemma 2.2.9. Let (V, T) be a complex representation of the compact Lie group T.

Let a be a non-trivial T-orbit and suppose that for some z G a, iz is tangent to a.

Then

(1) ly is tangent to a for all y G a.

(2) exp(it)a = a for all t G R.

Proof Define the T-equivariant vector field X on V by X(x) — ix. Since X is

tangent to a at z, it follows by T-equivariance that X is everywhere tangent to a,

proving (1). The flow $ of X is given by £(#, t) = exp(it)x. Statement (2) follows

since a is ^-invariant. •

2.3. Isotropy types for representations. Let (V, T) be a finite dimensional real

representation. It is well-known and elementary that the set 0(V, T) of isotropy

types is finite. If r, ji G (D, it follows from linearity and slice theory that

r ji if and only if Vr C 9V^

We say that an orbit type r is maximal (respectively, submaximal) if (i) r ^ (T)

and (ii) \x r implies \i = (T) (respectively, r ^ (r) and r is not maximal).

Given r G O, choose x G FT and let iV^r^) denote the normalizer of Tx in T.

Define

gT = dim(r • x), nT - dim(A^(rx)/rx)

We also define r

r

to be the rank of the identity component of N(TX)/TX. Of course,

gT, nr and r

r

depend only on r and not on the choice of x in VT.

2.4. Polynomial Invariants and Equivariants. Let P(V) denote the M-algebra

of R-valued polynomial functions on V and P(V, V) be the P(l/)-module of all

polynomial endomorphisms of V. We have

P(V) = 0 P\V) and P(F, V) = ©

Pfc(V,

V)

/c0 fc0

where

Pfc(y)

(respectively,

Pk(V,

V)) is the vector space of all homogeneous poly-

nomials (respectively, homogeneous polynomial maps) of degree k. For m G N, we

set

p("0(V)= 0 Pk(V)tmdP(m\V,V)= 0 Pk(V,V)

0km 0km

Suppose that (V, T) is a finite dimensional real representation. The action of T

on V extends to a linear action on the spaces P(V) and P(V, V) and all sums

of subspaces of homogeneous polynomial maps. For example, the action of T on

the space L(V,V) of real linear maps of V is given by 7 • A — 7Ay

- -1,

7 G T,

^4 G I/(V, V). The fixed point spaces of these actions define the spaces of invariant

and equivariant polynomial maps.

We let

P(V)r

denote the E-subalgebra of P(V) consisting of all T-invariant

polynomials and Pr(V, V') be the

P(F)r-module

of T-equivariant polynomial maps.

We have

P{V)r

= 0

f c o

P

f c

(^)

r

and Pr(V,V) =

(Bkopr(v v) w h e r e

^ ( ^ )

r

and

Pk(V,

V) are defined in the obvious way.

We say that a polynomial map R : V—V is radial if R — hly for some h G

P(V) (if (V, T) is a complex representation, we allow the invariant h to be complex

valued). We let i?(V, V) = ©^

0

Rk(V, V) be the graded linear subspace of P(V, V)