Stratified Kähler structures on adjoint quotients

Given a compact Lie group, endowed with a bi-invariant Riemannian metric, its complexification inherits a Kähler structure having twice the kinetic energy of the metric as its potential, and Kähler reduction with reference to the adjoint action yields a stratified Kähler structure on the resulting adjoint quotient. Exploiting classical invariant theory, in particular bisymmetric functions and variants thereof, we explore the singular Poisson–Kähler 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 Kähler quantization of the geodesic flow on the reduced level yields the irreducible algebraic characters of the complexified group.