Gelfand pairs associated with the action of graph automaton groups

Abstract

Graph automaton groups constitute a special class of automaton groups constructed from a graph. In this paper, we show that the action of any graph automaton group on each level of the rooted regular tree gives rise to a Gelfand pair. In particular, we determine the irreducible submodules of the action of such a group on the space of functions defined on each level of the tree, and we exhibit the corresponding spherical functions.

Keywords

• graph automaton group
• gelfand pair
• rooted regular tree
• $2$-points homogeneous action
• spherical function