Gelfand pairs and strong transitivity for Euclidean buildings

TL;DR

This paper establishes that for groups acting on Euclidean buildings, the Gelfand pair property combined with cocompactness implies strong transitivity, linking algebraic and geometric symmetries.

Contribution

It proves a converse to a known result, showing that Gelfand pairs and cocompact actions imply strong transitivity on Euclidean buildings.

Findings

Gelfand pair plus cocompactness implies strong transitivity

Existence of strongly regular hyperbolic elements is key

Equivalence between strong transitivity on the building and at infinity

Abstract

Let G be a locally compact group acting properly by type-preserving automorphisms on a locally finite thick Euclidean building $\Delta$ and K be the stabilizer of a special vertex in $\Delta$. It is known that (G, K) is a Gelfand pair as soon as G acts strongly transitively on $\Delta$; this is in particular the case when G is a semi-simple algebraic group over a local field. We show a converse to this statement, namely: if (G, K) is a Gelfand pair and G acts cocompactly on $\Delta$, then the action is strongly transitive. The proof uses the existence of strongly regular hyperbolic elements in G and their peculiar dynamics on the spherical building at infinity. Other equivalent formulations are also obtained, including the fact that G is strongly transitive on $\Delta$ if and only if it is strongly transitive on the spherical building at infinity.