Furstenberg boundary

Let G be a connected semisimple Lie group. The Furstenberg boundary of G is the homogeneous space :G/P, where P is a minimal parabolic subgroup of G. This space is compact and homogeneous under the action of G. More generally, quotients G/Q by parabolic subgroups Q are generalized flag manifolds, and the Furstenberg boundary is the maximal one among these in the sense that every quotient by a parabolic subgroup is a factor of G/P. For example, if G=\mathrm{SL}(n,\mathbb R), then the Furstenberg boundary is the manifold of complete flags in \mathbb R^n. For G=\mathrm{SL}(2,\mathbb R), it is \mathbb{RP}^1. == Relation to Poisson boundaries ==

Let \mu be a probability measure on G. A function f on G is called \mu-harmonic if :f(g)=\int_G f(gg')\,d\mu(g'). The Poisson boundary of the measured group (G,\mu) is a measure space that represents bounded \mu-harmonic functions by boundary integrals. Unlike the Furstenberg boundary, the Poisson boundary depends on the choice of the measure \mu. For semisimple Lie groups, Furstenberg showed that for broad classes of measures the Poisson boundary can be realized on a homogeneous boundary of the form G/Q, where Q is a parabolic subgroup. In particular situations the maximal boundary G/P plays the role of a universal homogeneous boundary from which the others are obtained as quotients. ==References==