The number of connected components in the double Bruhat cells for nonsimply-laced groups

Connected components of real double Bruhat cells

Double Bruhat cells in a semisimple group are intersections of cells in two Bruhat decompositions corresponding to two opposite Borel subgroups. They form a geometric framework for the study of total positivity in semisimple groups; they are also closely related to symplectic leaves in the corresponding Poisson-Lie groups. The term "cells" might be misleading because their topology can be quite non-trivial. As a first step towards understanding this topology, we enumerate the connected components of real double Bruhat cells. This result extends (from the simply-laced case to the general one) a…