Some translations in the second lowest two-sided cell of an affine Weyl group

TL;DR

This paper investigates specific translations within the second lowest two-sided cell of affine Weyl groups, refining Shi's conjecture on the number of left cells, and verifies the refinement for certain types.

Contribution

It identifies particular translations in the second lowest two-sided cell and proposes a refined conjecture, verified for types A_{n-1} and G_2.

Findings

Identified translations in the second lowest two-sided cell.

Formulated a refinement of Shi's conjecture.

Verified the refinement for types A_{n-1} and G_2.

Abstract

We are interested in cell properties of translations in affine Weyl groups. We find out some translations in the second lowest two-sided cell of an affine Weyl group and use the translations to formulate a refinement of Jianyi Shi's conjecture on the number of left cells in the second lowest two-sided cell. We verify the refinement for an affine Weyl group of type $\tilde A_{n-1}$ and type $\tilde G_2$.