The based ring the lowest generalized two-sided cell of an extended affine Weyl group

TL;DR

This paper determines the structure of the based ring of the lowest generalized two-sided cell in an extended affine Weyl group, verifies Lusztig's conjectures for this cell, and explores implications for Hecke algebras with unequal parameters.

Contribution

It explicitly describes the based ring structure of the lowest generalized two-sided cell and confirms Lusztig's conjectures P1-P15 for this cell, advancing understanding of affine Weyl groups.

Findings

Lusztig's conjectures P1-P15 hold for the lowest generalized two-sided cell.

The based ring of this cell is explicitly determined.

Certain simple modules of Hecke algebras are characterized using this structure.

Abstract

Let $\mathbf{c}_0$ be the lowest generalized two-sided cell of an extended affine Weyl group W. We determine the structure of the based ring of $\mathbf{c}_0$. For this we show that certain conjectures of Lusztig on generalized cells (called P1-P15) hold for $\mathbf{c}_0$. As an application, we use the structure of the based ring to study certain simple modules of Hecke algebras of $ W $ with unequal parameters, namely those attached to $\mathbf{c}_0$. Also we give a set of prime ideals $\mathfrak{p}$ of the center $\mathcal{Z}$ of the generic affine Hecke algebra $\mathcal{H}$ such that the reduced affine Hecke algebra $k_\mathfrak{p}\mathcal{H}$ is simple over $k_\mathfrak{p}$, where $k_\mathfrak{p}=\rm{Frac}(\mathcal{Z}/\mathfrak{p})$ is the residue field of $\mathcal{Z}$ at $\mathfrak{p}$. In particular, we show that the algebra $\mathcal{H}\otimes_\mathcal{Z}\rm{Frac}(\mathcal{Z})$ is a split simple algebra over the field $ \rm{Frac}(\mathcal{Z})$.