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 $G=\mathrm {SL} (2,\mathbb {R} )$, the Furstenberg boundary is the real projective line $\mathbb {RP} ^{1}$, 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.

Semisimple Lie groups

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

• Borel, Armand; Ji, Lizhen, Compactifications of symmetric and locally symmetric spaces (PDF)
• Furstenberg, Harry (1963), "A Poisson Formula for Semi-Simple Lie Groups", Annals of Mathematics, 77 (2): 335–386, doi:10.2307/1970220, JSTOR 1970220
• Furstenberg, Harry (1973), "Boundary theory and stochastic processes on homogeneous spaces", in Calvin Moore (ed.), Harmonic Analysis on Homogeneous Spaces, Proceedings of Symposia in Pure Mathematics, vol. 26, AMS, pp. 193–232, doi:10.1090/pspum/026/0352328, ISBN 9780821814260