On $J$-folded alcove paths and combinatorial representations of affine Hecke algebras

TL;DR

This paper introduces a combinatorial model of J-folded alcove paths in affine Weyl groups to construct and analyze representations of affine Hecke algebras, linking to Kazhdan-Lusztig theory and Opdam's Plancherel Theorem.

Contribution

It presents a novel combinatorial framework for affine Hecke algebra representations and explores their properties and connections to existing theories.

Findings

Constructed representations using J-folded alcove paths

Studied boundedness of these representations

Proposed conjectures linking to Kazhdan-Lusztig theory and Plancherel Theorem

Abstract

We introduce the combinatorial model of $J$-folded alcove paths in an affine Weyl group and construct representations of affine Hecke algebras using this model. We study boundedness of these representations, and we state conjectures linking our combinatorial formulae to Kazhdan-Lusztig theory and Opdam's Plancherel Theorem.