Many classical locally compact groups $G$ admit a very large compact subgroup $K$, where "very large" has been formalized by Gelfand in 1950. Examples include $G=\mathrm{SL}_n(\mathbb{R})$ with $K=\mathrm{SO}(n)$, or $G=\mathrm{SL}_n(\mathbb{Q}_p)$ with $K=\mathrm{SL}_n(\mathbb{Z}_p)$. More generally, all semi-simple algebraic groups and some tree automorphism groups.

In these explicit examples, there is also an "Iwasawa decomposition" which formalizes the fact that $G$ has a homogeneous Frustenberg boundary, even homogeneous under $K$. This is a very strong restriction for general groups.

Using no structure theory whatsoever, we prove that this homogeneity (and Iwasawa decomposition) holds for all Gelfand Pairs. This implies, in some geometric cases, a classification of Gelfand pairs. (This is related to a small part of my 2021 zoom colloquium at UCSD).