Furstenberg boundary

In potential theory, a discipline within applied mathematics, 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, roughly speaking, is a universal moduli space for the Poisson integral, expressing a harmonic function on a group in terms of its boundary values.

Motivation

A model for the Furstenberg boundary is the hyperbolic disc ${\displaystyle D=\{z:|z|<1\}}$. The classical Poisson formula for a bounded harmonic function on the disc has the form

${\displaystyle f(z)={\frac {1}{2\pi }}\int _{0}^{2\pi }{\hat {f}}(e^{i\theta })P(z,e^{i\theta })\,d\theta }$

where P is the Poisson kernel. Any function f on the disc determines a function on the group of Möbius transformations of the disc by setting F(g) = f(g(0)). Then the Poisson formula has the form

${\displaystyle F(g)=\int _{|z|=1}{\hat {f}}(gz)\,dm(z)}$

where m is the Haar measure on the boundary. This function is then harmonic in the sense that it satisfies the mean-value property with respect to a measure on the Möbius group induced from the usual Lebesgue measure of the disc, suitably normalized. The association of a bounded harmonic function to an (essentially) bounded function on the boundary is one-to-one.

Construction for semi-simple groups

In general, let G be a semi-simple Lie group and μ a probability measure on G that is absolutely continuous. A function f on G is μ-harmonic if it satisfies the mean value property with respect to the measure μ:

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

There is then a compact space Π, with a G action and measure ν, such that any bounded harmonic function on G is given by

${\displaystyle f(g)=\int _{\Pi }{\hat {f}}(gp)\,d\nu (p)}$

for some bounded function ${\displaystyle {\hat {f}}}$ on Π.

The space Π and measure ν depend on the measure μ (and so, what precisely constitutes a harmonic function). However, it turns out that although there are many possibilities for the measure ν (which always depends genuinely on μ), there are only a finite number of spaces Π (up to isomorphism): these are homogeneous spaces of G that are quotients of G by some parabolic subgroup, which can be described completely in terms of root data and a given Iwasawa decomposition. Moreover, there is a maximal such space, with quotient maps going down to all of the other spaces, that is called the Furstenberg boundary.

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
• Furstenberg, Harry (1973), Calvin Moore (ed.), "Boundary theory and stochastic processes on homogeneous spaces", Proceedings of Symposia in Pure Mathematics, AMS, 26: 193–232