The elliptic Calogero-Moser Hamiltonian and Lax pair associated with a
general simple Lie algebra G are shown to scale to the (affine) Toda
Hamiltonian and Lax pair. The limit consists in taking the elliptic mo
dulus tau and the Calogero-Moser couplings m to infinity, while keepin
g fixed the combination M = m e(i pi delta tau) for some exponent delt
a. Critical scaling limits arise when 1/delta equals the-Coxeter numbe
r or the dual Coxeter number for the untwisted and twisted Calogero-Mo
ser systems respectively; the limit consists then of the Toda system f
or the affine Lie algebras G((1)) and (G((1)))(boolean OR). The limits
of the untwisted or twisted Calogero-Moser system, for delta less tha
n these critical values, but non-zero, consists of the ordinary Toda s
ystem, while for delta = 0, it consists of the trigonometric Calogero-
Moser systems for the algebras G and G(boolean OR) respectively. (C) 1
998 Published by Elsevier Science B.V.