The based rings of two-sided cells in an affine Weyl group of type $\tilde B_3$, II

TL;DR

This paper computes the based rings of two-sided cells in an affine Weyl group of type B_3, verifying Lusztig's conjecture for some cells and suggesting modifications for others, advancing understanding of algebraic structures related to unipotent classes.

Contribution

It explicitly calculates based rings for specific two-sided cells in B_3 and tests Lusztig's conjecture, providing new insights and potential revisions.

Findings

Verification of Lusztig's conjecture for two cells

Partial evidence for modification of the conjecture

Enhanced understanding of based rings in affine Weyl groups

Abstract

We compute the based rings of two-sided cells corresponding to the unipotent classes in $Sp_6(\mathbb C)$ with Jordan blocks (33), (411), (222) respectively. The results for the first two two-sided cells also verify Lusztig's conjecture on the structure of the based rings of two-sided cells of an affine Weyl group. The result for the last two-sided cell partially suggests a modification of Lusztig's conjecture on the structure of the based rings of two-sided cells of an affine Weyl group.