Furstenberg boundary

In mathematics, specifically harmonic analysis and probability theory, the Furstenberg boundary is a notion of boundary associated with a group. It is named for Harry Furstenberg, who introduced it in a series of papers beginning in 1963 (in the case of semisimple Lie groups). The Furstenberg boundary can be characterized as a universal boundary space for harmonic analysis on the group, in the sense that bounded harmonic functions can be represented by their boundary values via a Poisson-type integral.

For example, when ${\displaystyle G=\mathrm {SL} (2,\mathbb {R} )}$, the Furstenberg boundary is the real projective line ${\displaystyle \mathbb {RP} ^{1}}$, which may be identified with the boundary circle of the hyperbolic plane, and the Poisson-like integral is the usual Poisson kernel for the upper half-plane.

Let ${\displaystyle G}$ be a connected semisimple Lie group. The Furstenberg boundary of ${\displaystyle G}$ is the homogeneous space

$${\displaystyle G/P,}$$

where ${\displaystyle P}$ is a minimal parabolic subgroup of ${\displaystyle G}$.

This space is compact and homogeneous under the action of ${\displaystyle G}$. More generally, quotients ${\displaystyle G/Q}$ by parabolic subgroups ${\displaystyle Q}$ 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 ${\displaystyle G/P}$.

For example, if ${\displaystyle G=\mathrm {SL} (n,\mathbb {R} )}$, then the Furstenberg boundary is the manifold of complete flags in ${\displaystyle \mathbb {R} ^{n}}$. For ${\displaystyle G=\mathrm {SL} (2,\mathbb {R} )}$, it is ${\displaystyle \mathbb {RP} ^{1}}$.

Let ${\displaystyle \mu }$ be a probability measure on ${\displaystyle G}$. A function ${\displaystyle f}$ on ${\displaystyle G}$ is called ${\displaystyle \mu }$-harmonic if

$${\displaystyle f(g)=\int _{G}f(gg')\,d\mu (g').}$$

The Poisson boundary of the measured group ${\displaystyle (G,\mu )}$ is a measure space that represents bounded ${\displaystyle \mu }$-harmonic functions by boundary integrals. Unlike the Furstenberg boundary, the Poisson boundary depends on the choice of the measure ${\displaystyle \mu }$.

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 ${\displaystyle G/Q}$, where ${\displaystyle Q}$ is a parabolic subgroup. In particular situations the maximal boundary ${\displaystyle G/P}$ plays the role of a universal homogeneous boundary from which the others are obtained as quotients.