The Based Rings of Two-sided cells in an Affine Weyl group of type $\tilde B_3$, IV
Yannan Qiu
TL;DR
The paper determines the based ring $J_c$ of the two-sided cell in the extended affine Weyl group of type $\tilde{B}_3$ corresponding to the unipotent class with Jordan blocks $(21111)$ in $Sp_6(\mathbb C)$, where $a(c)=6$. By leveraging Du's cell decomposition and Lusztig's asymptotic Hecke algebra framework, it constructs an explicit bijection $\pi$ from $c$ to irreducible $F_c$-vector bundles on $Y\times Y$ and shows $J_c\cong K_{F_c}(Y\times Y)$ with $F_c=Sp_4(\mathbb C)\times \mathbb Z/2\mathbb Z$. The work further identifies left-cell pieces with irreducible representations of $F_c$, establishing compatible bimodule structures across the 24 left cells and thereby confirming Lusztig's conjecture for this cell. This explicit realization enables practical computation of irreducible representations of affine Hecke algebras of type $\tilde{B}_3$ and strengthens the geometric understanding of $J_c$ in a nontrivial affine setting.
Abstract
We compute the based ring of two-sided cell corresponding to the unipotent class in $Sp_6(\mathbb C)$ with Jordan blocks (21111). The results also verify Lusztig's conjecture on the structure of the based rings of the two-sided cells of an affine Weyl group.