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).