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 {{math|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 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:

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

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

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

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.

References

{{citation|first1=Armand|last1=Borel|first2=Lizhen|last2=Ji|title=Compactifications of symmetric and locally symmetric spaces|url=http://dept.math.lsa.umich.edu/~lji/head.pdf}}
{{citation|first=Harry|last=Furstenberg|title=A Poisson Formula for Semi-Simple Lie Groups|journal=Annals of Mathematics|volume=77|issue=2|year=1963|pages=335–386|doi=10.2307/1970220|jstor=1970220}}
{{citation|first=Harry|last=Furstenberg|title=Boundary theory and stochastic processes on homogeneous spaces|publisher=AMS|year=1973|pages=193–232|editor=Calvin Moore|journal=Proceedings of Symposia in Pure Mathematics|volume=26|doi=10.1090/pspum/026/0352328|isbn=9780821814260}}

Category:Potential theory