Furstenberg boundary explained

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

D=\{z:|z|<1\}

. The classical Poisson formula for a bounded harmonic function on the disc has the form

f(z)=

	1
	2\pi

	2\pi
\int
	0

\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 . 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 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:

f(g)=\int_Gf(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

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

for some bounded function

\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.