On calibrated representations and the Plancherel Theorem for affine Hecke algebras

TL;DR

This paper extends Ram's construction of calibrated representations to multi-parameter affine Hecke algebras and derives the Plancherel formulae for low-rank cases, providing explicit representation constructions.

Contribution

It generalizes the explicit construction of calibrated representations to multi-parameter cases and derives Plancherel formulae for rank 1 and 2 affine Hecke algebras.

Findings

Generalized Ram's construction to multi-parameter affine Hecke algebras

Derived Plancherel formulae for rank 1 and 2 cases

Constructed all representations involved in the Plancherel formulae

Abstract

This paper has two main purposes. Firstly we generalise Ram's explicit construction of calibrated representations of the affine Hecke algebra to the multi-parameter case (including the non-reduced $BC_n$ case). We then derive the Plancherel formulae for all rank~1 and rank~2 affine Hecke algebras including a construction of all representations involved (following work of Opdam).