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 $$D=\{z : |z|<1\}$$. The classical Poisson formula for a bounded harmonic function on the disc has the form
 * $$f(z) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(e^{i\theta}) P(z,e^{i\theta})\, d\theta$$

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


 * $$F(g) = \int_{|z|=1}\hat{f}(gz) \, dm(z)$$

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 &mu; a probability measure on G that is absolutely continuous. A function f on G is &mu;-harmonic if it satisfies the mean value property with respect to the measure &mu;:


 * $$f(g) = \int_G f(gg') \, d\mu(g')$$

There is then a compact space &Pi;, with a G action and measure &nu;, such that any bounded harmonic function on G is given by


 * $$f(g) = \int_\Pi \hat{f}(gp) \, d\nu(p)$$

for some bounded function $$\hat{f}$$ on &Pi;.

The space &Pi; and measure &nu; depend on the measure &mu; (and so, what precisely constitutes a harmonic function). However, it turns out that although there are many possibilities for the measure &nu; (which always depends genuinely on &mu;), there are only a finite number of spaces &Pi; (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.