Furstenberg boundary of a semisimple Lie group

Definition

Let $G$ be a connected noncompact real semisimple Lie group with finite center, and let $P$ be a minimal parabolic subgroup. The Furstenberg boundary is the compact homogeneous space

\partial_F G=G/P.

Different minimal parabolics are conjugate, so the resulting $G$ -space is well defined up to $G$ -equivariant diffeomorphism. If $P=MAN$ is its Langlands decomposition and $K$ is the corresponding maximal compact subgroup, the Iwasawa decomposition induces a $K$ -equivariant identification $G/P\cong K/M$ .

Geometric interpretation

The space $G/P$ is also called the minimal real flag manifold. For $G=\operatorname{SL}(n,\mathbb R)$ , it is the space of complete flags in $\mathbb R^n$ . For $G=\operatorname{SL}(2,\mathbb R)$ , it is $\mathbb{RP}^1$ , which is a circle. Compactness follows from $G=KP$ , while transitivity is built into the quotient construction Helgason, Chapter I.

Role in representation theory

Functions or sections of equivariant bundles over $G/P$ provide compact models of spherical and more general principal-series representations. Boundary values of eigenfunctions on the symmetric space $G/K$ likewise live naturally on $G/P$ . The identification $G/P\cong K/M$ makes the compact-group action available without discarding the full $G$ -action.

Terminology and scope

References

Sigurdur Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Mathematical Surveys and Monographs 83, American Mathematical Society, 2000. AMS record. Relevant: Chapter I on $G/K$ , Iwasawa decomposition, and the boundary $K/M$ .
Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton Mathematical Series 36, Princeton University Press, 1986. Author-maintained record. Relevant: Chapter VII on principal-series representations and their compact picture.