2. TECHNICAL PRELIMINARIES AND BASIC NOTATIONS

2.1. T-sets and isotropy types. Let T be a group and X be a T-set. If x G T,

then T • x denotes the T-orbit of x and Tx denotes the isotropy subgroup of T at

x. Let (Tx) denote the conjugacy class of the subgroup Tx in T. We say that (Tx)

is the isotropy type or orbit type of x. Let 0(X, T) denote the set of isotropy types

for the T-set X. We frequently abbreviate 0(X,T) to O if X and T are implicit

from the context. For x G T, let L(X) denote the isotropy type of x. If r G 0(X1 T),

let XT = {x G X | t(x) = r} be the set of points of isotropy type r. Let X

r

denote

the fixed point set for the action of T on X. We define the usual partial order for

bifurcation theory on 0(X, T) by "r /x if there exists H G T, K G // such that

2.2. Representations. Suppose that V is a compact Lie group. Let F denote

either the field of real numbers IR or the field of complex numbers C. Let V be a

(finite dimensional) vector space over F. We say that (V, V) is an F-representation

if T acts on V as a group of F-linear transformations. We refer to V as a real or

complex representation space for V according as to whether F = R or F = C.

Let V be a nontrivial (finite-dimensional) real representation space for V. Av-

eraging a positive definite inner product for V over Y using Haar measure we may

suppose that V has a positive definite T-invariant inner product (,) with asso-

ciated norm | • |. In particular, we may regard Y as acting on V by orthogonal

transformations.

Suppose instead that V is a nontrivial (finite-dimensional) complex representa-

tion space for I\ As above, we may assume that V has a positive definite T-invariant

hermitian inner product , . In particular, we may regard (V, Y) as a unitary rep-

resentation. If we let (,) denote the real part of , , then (,) is a T-invariant

inner product on V. Let Jy denote the complex structure on V defined by scalar

multiplication by i. That is, Jy(v) — iv, all v G V. If we regard S1 C C as the

group of complex numbers of unit modulus, we may extend the action of T on V

to an action of V x S1 on V. The resulting representation (V, T x S'1) is complex

and both , and (,) will be F x

S1

-invariant. In particular, we have

(4) (v, Jvv) = 0, (tiE V)

In the sequel, we always assume that the T-invariant inner product on V satisfies (4).

We always use the notation S1 for the group of complex numbers of unit modulus

and take the 51-action on V defined by scalar multiplication.

Suppose that (V, T) is a real representation. Let Lr(V, V) denote the linear space

of all T-equivariant E-linear maps from V to V.

Definition 2.2.1. Let (V, T) be a nontrivial irreducible T-representation. We say

that

(1) (V, T) is absolutely irreducible if £,r(V, V) ^ R.

(2) (V, T) is irreducible of complex type if Lr(V, V) = C.

(3) (V, r) is irreducible of quaternionic type if Lr(V, V) = H (the quaternions).

Remarks 2.2.2. (1) By Frobenius' Theorem [51, 7.7, p.430], every real irreducible

representation (V, V) is either absolutely irreducible, irreducible of complex type

or irreducible of quaternionic type. In this work, we shall not explicitly consider

examples where (V,r) is irreducible of quaternionic type. (See [19, §3] for an

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