C*-algebraic intertwiners for principal series: case of SL(2)

TL;DR

This paper constructs and normalizes intertwining operators for the principal series of SL(2) using Fourier transforms on homogeneous spaces, providing explicit normalizing factors and relating reducibility points to specific distributions.

Contribution

It introduces a C*-algebraic framework for intertwiners of SL(2) principal series, with explicit normalization and connection to reducibility points.

Findings

Explicit normalizing factors for intertwiners are derived.

Normalisation uses Fourier transform twisted by a Weyl element.

Reducibility points are linked to a specific distribution.

Abstract

We construct and normalise intertwining operators at the level of Hilbert modules describing the principal series of SL(2). Normalisation is achieved through the use of a Fourier transform defined on some homogenous space and twisted by a Weyl element. Normalising factors are also explicitely obtained. In the appendix we relate reducibility points to a certain distribution arising from the non-normalised intertwiners.