Connected components of real double Bruhat cells

Abstract

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 nontrivial. 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) and proves the conjecture made in a joint work with B. Shapiro, M. Shapiro, and A. Vainstein; it also extends earlier work by B. Shapiro, M. Shapiro, A. Vainstein, and K. Rietsch.