Let K be a compact, connected Lie group and K-C its complexification.
I consider the Hilbert space HL(2)(K-C, v(t)) of holomorphic functions
introduced in [H1], where the parameter t is to be interpreted as Pla
nck's constant. In Light of CL-S], the complex group K-C may be identi
fied canonically with the cotangent bundle of K. Using this identifica
tion I associate to each F epsilon HL(2)(K-C, v(t)) a ''phase space pr
obability density.'' The main result of this paper is Theorem 1, which
provides an upper bound on this density which holds uniformly over al
l F and all points in phase space. Specifically, the phase space proba
bility density is at most a(t)(2 pi t)(-n), where n = dim K and a(t) i
s a constant which tends to one exponentially fast as t tends to zero.
At least for small t, this bound cannot be significantly improved. Wi
th t regarded as Planck's constant, the quantity (2 pi t)(-n) is preci
sely what is expected on physical grounds. Theorem 1 should be interpr
eted as a form of the Heisenberg uncertainty principle for K, that is,
a limit on the concentration of states in phase space. The theorem su
pports the interpretation of the Hilbert space HL(2)(K-C, v(t)) as the
phase space representation of quantum mechanics for a particle with c
onfiguration space K. The phase space bound is deduced from very sharp
pointwise bounds on functions in HL(2)(K-C, v(t)) (Theorem 2). The pr
oofs rely on precise calculations involving the heat kernel on K and t
he heat kernel on K-C/K.