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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), Calvin Moore (ed.), “Boundary theory and stochastic processes on homogeneous spaces”, Proceedings of Symposia in Pure Mathematics, 26, AMS: 193–232, doi:10.1090/pspum/026/0352328, ISBN 9780821814260{{citation}}: CS1 maint: work parameter with ISBN (link)

Sample Page

Semisimple Lie groups

Relation to Poisson boundaries

References