A natural extension of Kirillov-Kostant orbit method relates geometry of Poisson-Lie
groups to representation theory of Hopf algebras. All compatible Poisson structures
on complex semisimple Lie groups were classified by Belavin and Drinfeld in the
early 80's. Very little is known about the geometry of these structures except for the
special case of the so called standard structure.
We will describe a classification of the symplectic leaves of all Poisson-Lie groups
of Belavin and Drinfeld. It is connected to interesting combinatorics of minimal
length representatives of cosets in Coxeter groups and is ultimately related to works
of Duflo and Moeglin-Rentschler on primitive spectrum of Universal enveloping
algebras of non-semisimple Lie algebras.
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