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
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