Analytic R-groups of affine Hecke algebras

TL;DR

This paper introduces analytic R-groups for affine Hecke algebras, proves a key dimension theorem, and characterizes the commutant algebra of unitary principal series representations, revealing trivial cocycles in classical cases.

Contribution

It defines analytic R-groups for affine Hecke algebras and establishes their properties, including a dimension theorem and the structure of commutant algebras.

Findings

Proved the analog of the Knapp-Stein Dimension Theorem for affine Hecke algebras.

Showed the commutant algebra of a unitary principal series is isomorphic to a twisted group algebra.

Established that the twisting cocycle is trivial for classical Hecke algebras.

Abstract

We define analytic $R$-groups for affine Hecke algebras, and prove the analog of the Knapp-Stein Dimension Theorem. As a corollary we prove that the commutant algebra of a unitary principal series representation is isomorphic to the complex group algebra of the $R$-group, twisted by a certain 2-cocycle $\gamma$. For classical Hecke algebras we prove that $\gamma$ is always trivial.