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

TL;DR

This paper investigates the structure of based rings in affine Weyl groups of type B_3, revealing that Lusztig's conjecture requires modification for certain two-sided cells related to specific unipotent classes.

Contribution

It identifies necessary modifications to Lusztig's conjecture concerning based rings in affine B_3, advancing understanding of their algebraic structure.

Findings

Lusztig's conjecture does not hold in its original form for certain two-sided cells in B_3

The structure of based rings must be adjusted for the unipotent class with 3 equal Jordan blocks

Provides new insights into the algebraic properties of affine Weyl groups of type B_3

Abstract

For type $\tilde B_3$ we show that Lusztig's conjecture on the structure of the based ring of two-sided cell corresponding to the unipotent class in $Sp_6(\mathbb C)$ with 3 equal Jordan blocks needs modified.