文摘
Given a compact Lie group, endowed with a bi-invariant Riemannian metric, its complexification inherits a Kxe4;hler structure having twice the kinetic energy of the metric as its potential, and Kxe4;hler reduction with reference to the adjoint action yields a stratified Kxe4;hler structure on the resulting adjoint quotient. Exploiting classical invariant theory, in particular bisymmetric functions and variants thereof, we explore the singular Poisson–Kxe4;hler 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 Kxe4;hler quantization of the geodesic flow on the reduced level yields the irreducible algebraic characters of the complexified group.