CHAPTE R 2

Action-Group Coordinates

In this section we describe the action-group model space. Chap. 3 will state

conditions under which this model space is realizable in a particular system, and

give a brief historical sketch of its origins.

Let G denote a (real-analytic) compact connected Lie group and let Gc denote

its complexification (see, e.g., Brocker and torn Dieck (1985)). For computing

estimates later on we need to assume that G is realized as a real-analytic subgroup

of SO (no, M) for some integer UQ- Since G is compact and connected this is always

possible, by a corollary of the Peter-Weyl theorem (see, e.g., op. cit, Theorem 4.1

and Exercise 4.7.1). We may then identify the Lie algebra of G with a subalgebra

Q C ]RnGXnG The adjoint action can be written as Ad^ £ = g£g~l, and the Lie

bracket as [£1,^2] = ad ^ £2 = £162 ~~ £2^1 • The complexification Gc of G can be

identified with a complex subgroup of SO(riG,C). The Lie algebra $c of Gc is

identifiable with the complex subalgebra g 0 ig C C n c X n G .

Henceforth T C G denotes a fixed maximal torus, t its Lie algebra,

and W c t a Weyl chamber in g*.

T h e mode l space and its symplectic structure

The natural projection g* — » t* (the dual map of inclusion) has kernel t 1 and

thus restricts, by virtue of 1.4, to an isomorphism i : t — » t*. This map identifies

W with an open set tg = i(W) C t*. One calls 1Q a Weyl Chamber also.

We define the action-group model space for a compact connected Lie group G

as G x tj . A natural symplectic structure UJQ on G x tg? generalizing the canonical

structure V dQj A dpj on T n x R n , is given by the following proposition:

2.1 Proposition . Equip T*G with its natural symplectic structure. For each a G

0*, let ac denote the left-invariant one-form on G with ac^id^ ) = OL (here viewing

one-forms as sections of the cotangent bundle). Then the embedding G x tg ^ T*G

that maps (g,p) to {i~l(p))G(g)j maps G x tj onto a symplectic submanifold of

T*G.

We define UJQ to be the symplectic structure on G x tg pulled back by this

embedding. Note that UJQ does not depend on the choice of Ad-invariant inner

product on 9, or on the realization of G as a linear group. Proposition 2.1 follows

from a more general observation we shall make in Part 2, or may be verified directly.

The injection Lp~x o z _ 1 : t* ^ Q maps t* isomorphically onto t (see 1.5), and

so identifies tg with a Weyl chamber tg C t in g. For computing estimates later

on, it is convenient for us to make the corresponding identification of G x tg with

G x to. We now describe explicitly the symplectic structure on G x tg induced by

this identification. This symplectic structure of course does depend on the choice

of Ad-invariant inner product. Since we eventually need a complexification of the

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