Irreducible representations of tree automorphism groups into Pontryagin spaces

Abstract

Let G = Aut(T) be the automorphism group of a regular tree T. We study continuous irreducible representations of G that preserve a continuous strongly nondegenerate sesquilinear form of finite index on a Hilbert space. These are already classified for index 0 (unitary case) and for index 1. We show that there are no more representations for index > 1, which completes the classification.

arXiv commentv2: we have proved the result for continuous representations into Pontryagin spaces, adding an appendix where we show admissibility of such representations. v3: Final version, published in Geometriae Dedicata. Corrected a few minor errors and improved the exposition. Added a statement declaring minor use of AI, limited to the later stages of preparation and revision of the paper. 43 pages