Analogs of principal series representations for Thompson's groups $F$ and $T$

TL;DR

This paper introduces new series of irreducible representations for Thompson's groups $F$ and $T$, classifies them, and distinguishes them from previously known induced representations.

Contribution

It defines analogs of principal series representations for Thompson's groups and provides a classification and distinction from induced representations.

Findings

Representations are irreducible

Classified up to unitary equivalence

Distinct from induced representations from stabilizers

Abstract

We define series of representations of the Thompson's groups $F$ and $T$, which are analogs of principal series representations of $SL(2,\R)$. We show that they are irreducible and classify them up to unitary equivalence. We also prove that they are different from representations induced from finite-dimensional representations of stabilizers of points under natural actions of $F$ and $T$ on the unit interval and the unit circle, respectively.