  Given a compact Lie group, endowed with a bi-invariant Riemannian metric, its
complexification inherits a Kaehler structure having twice the kinetic energy
of the metric as its potential, and Kaehler reduction with reference to the
adjoint action yields a stratified Kaehler structure on the resulting adjoint
quotient. Exploiting classical invariant theory, in particular bisymmetric
functions and variants thereof, we explore the singular Poisson-Kaehler
geometry of this quotient. Among other things we prove that, for various
compact groups, the real coordinate ring of the adjoint quotient is generated,
as a Poisson algebra, by the real and imaginary parts of the fundamental
characters. We also show that singular Kaehler quantization of the geodesic
flow on the reduced level yields the irreducible algebraic characters of the
complexified group.
