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.
(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.
gT = dim(r • x), nT - dim(A^(rx)/rx)
We also define r
to be the rank of the identity component of N(TX)/TX. Of course,
gT, nr and 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) = ©
V)) is the vector space of all homogeneous poly-
nomials (respectively, homogeneous polynomial maps) of degree k. For m G N, we
p("0(V)= 0 Pk(V)tmdP(m\V,V)= 0 Pk(V,V)
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
7 G T,
^4 G I/(V, V). The fixed point spaces of these actions define the spaces of invariant
and equivariant polynomial maps.
denote the E-subalgebra of P(V) consisting of all T-invariant
polynomials and Pr(V, V') be the
of T-equivariant polynomial maps.
f c o
and Pr(V,V) =
(Bkopr(v v) w h e r e
^ ( ^ )
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) = ©^
Rk(V, V) be the graded linear subspace of P(V, V)