x NOTATION

a projection

gc

— » t

c

along t

x c

(p. 12);

a1-

projection

gc

— • t^

c

along t

c

(p. 12)

a* projection g* —• » t along

t"1

(p. 56)

£ element of so(3, C) defined through

£u = £x u, for

^ G C 3

(p. 13)

{u, v} Poisson bracket of u and i; (p. 13)

£M infinitesimal generator of action of G on M,

along some ( e g

P symplectic manifold

J momentum map of G-action on P

J^ (-component of J (p. 15)

JG

momentum map of G-action on G x to (p. 16);

in Part 2, momentum map of G-action on G x tj (p. 55)

j

G

momentum map of T-action on G x to (p. 22);

in Part 2, momentum map of T-action on G x tj (p. 55)

ft frequency (=V/z) (p. 21)

IT integral lattice of torus T in t (p. 22)

H T-average of H (p. 23)

int M interior of M

Av(M)

space of continuous maps u : M — V that are real-analytic

(resp. holomorphic) on int M, for a real (resp. complex)

vector space V (p. 26)

|| • || supremum norm on

Av

(M) (p. 26)

closed i?-ball in t with center p (p. 27)

closed p-neighborhood of BR(P) in

tc(p.

27)

Hamiltonian analyticity widths (p. 27)

fixed element of t0 (or U C t0 (pp. 27,42)

variable parameters (p* G int 5 ) (p. 28)

shorthand for Bp(p) and Bp(jp) (see above)

the domain G^ x

B?r/(p*)

(p. 28)

BR(P)

BpRip)

a, /?

P

B, BP

7

Dy{P*,P)

C i , . . .

E

\-\s

fel,fe2,

a i , . . .

ft

Q l ,

. . .

n^-

c

7To

7TW

,c7

^ 3

ak

)

supremum norm on Ac""(p*,r)) (D1V (p. 28)

r/9

time constants (p. 29)

dimensionless constants (pp. 28-30)

energy constant (= ap/T^j) (p. 30)

norm on C

n c X n G

induced by Schur-Hadamard

product S (p. 32)

dimensionless constants (pp. 32,36)

number of positive roots of g in t* (p. 32)

positive real roots of g in t* (p. 32)

isomorphism t — t* induced by Killing form (p. 33)

inner product on t* induced by Killing form (p. 33)

inverse roots g in t (p. 33)

Cartan integers of g (p. 33)

see p. 33

dimensionless constant (p. 39)

set of points in M not in Z

projection g*eg — O {O a co-adjoint orbit; p. 47)

projection g*eg - W (p. 47)