HAMILTONIAN VECTO R FIELDS IN ACTION-GROUP COORDINATES 11

model space, we develop notation which also makes sense on the complex manifold

Gc x t c containing G x t0 (t c = t © it). For £0 G g c , r

0

G t c and (g,p) e Gc x t c ,

define the (complex) vector {£,o^o)gp tangent to Gc x t c at (g,p) by

2.2 (fo,T0),,P = ^ ( ^ e x p ( ^

0

) , P + tr

0

) | .

Note that every tangent vector in T(^

p

)(G c x t c ) is of this form.

On Gc x t c define the one-form 9 ^ by

2.3 (©G,(^o,ro)^,p) =p-^o ,

and define the two-form UJQ on G c x t c by

2.4

CJG

= -dOG .

We claim:

Restricted to G x to C Gc x tc, UJG agrees with the (real) symplectic

structure LUQ on G x tg defined above, after making the identification

G x t0 = G x tg discussed above.

For a proof, seee Lemma 9.4, Part 2. The above formulas also appear in Dazord and

Delzant (1987, Section 5) (who obtain it via a different route). For our applications

to perturbation theory, we need explicit equations of motion and an explicit formula

for the Poisson bracket. We turn to these next.

Hamiltonia n vector fields in action-group coordinates

If £ : Gc x t c —* Qc and r : Gc x t c —• t c are arbitrary holomorphic maps, we

JL

dg

define vector fields £ • •§- and r • -§- on Gc x tc by

£• | ^ (fl,p) = ($(7,p),0)g,p , ( T ' ^ ) (5.P) = (0,r(5,p))

9 l P

.

An arbitrary vector field on Gc x t c is then of the form £ • -^- + r • ^- for some

vector-valued functions £ : Gc x t € — g c and r : G c x t c -^ t c .

Write

t±c

=

t1-

©

it-1

and define tg to be the connected component containing

to of the set

{p e f | ad

p

: t ± c - • t ± c is invertible } .

For p G tg, let Ap : t ± c — » t ± c denote the inverse of ad

p

: t ± c — t x c . In particular,

let us record that

2-5 Ap([p,£]) = £=[p,AP(0] (eet i C ,p€tS)

and

2.6 t0 = tflt

0

:

.

It is possible to show that the restriction of the (complex) form UJQ to Gc x tg

is non-degenerate. If H : G c x tg —• C is holomorphic, then the corresponding

(complex) Hamiltonian vector field XH is defined by XH -I ^G = dH. Indeed

2-7

X H

= ^ H - ^ - + TH-^-

dg dp