Affine Hecke Algebras via DAHA

TL;DR

The paper introduces a new method to derive the Plancherel measure for Affine Hecke Algebras using limits of integral formulas from Double Affine Hecke Algebras, extending residue techniques.

Contribution

It provides a novel approach to compute the Plancherel measure for Affine Hecke Algebras via a limiting process involving integral formulas and residue methods.

Findings

Derived a limit formula for Plancherel measure

Extended residue techniques to affine settings

Presented explicit formulas for A_1 case

Abstract

A method is suggested for obtaining the Plancherel measure for Affine Hecke Algebras as a limit of integral-type formulas for inner products in the polynomial and related modules of Double Affine Hecke Algebras. The analytic continuation necessary here is a generalization of "picking up residues" due to Arthur, Heckman, Opdam and others, which can be traced back to Hermann Weyl. Generally, it is a finite sum of integrals over double affine residual subtori; a complete formula is presented for $A_1$ in the spherical case.