Cellularity of the lowest two-sided ideal of an affine Hecke algebra

TL;DR

This paper proves that the lowest two-sided ideal of an affine Hecke algebra is affine cellular for all parameters, providing explicit bases and combinatorial descriptions in type A.

Contribution

It establishes the affine cellularity of the lowest two-sided ideal in affine Hecke algebras and describes the cellular basis explicitly, including combinatorial insights in type A.

Findings

The lowest two-sided ideal is affine cellular for all parameters.

Explicit cellular basis with a nice decomposition in terms of Kazhdan-Lusztig basis.

Combinatorial description of the decomposition in type A using paths.

Abstract

In this paper we show that the lowest two-sided ideal of an affine Hecke algebra is affine cellular for all choices of parameters. We explicitely describe the cellular basis and we show that the basis elements have a nice decomposition when expressed in the Kazhdan-Lusztig basis. In type $A$ we provide a combinatorial description of this decomposition in term of number of paths.