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

We define series of representations of the Thompson’s groups $F$ and $T$, which are analogs of principal series representations of $SL(2,\mathbb{R})$. 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.