Lusztig's asymptotic algebra and a cell factorisation for the second lowest two-sided cell in type $\tilde{B}_n$

Abstract

We determine Lusztig's asymptotic algebra for the second lowest two-sided cell~$Δ_1$ in the equal parameter extended affine Weyl group $\tilde{\mathsf{B}}_n$ for all $n\geq 2$, realising it as a full matrix algebra over the representation ring of $\mathsf{Sp}_{2n-2}(\CC)\times C_2$ with $C_2$ the group of order~$2$. Our result is an explicit isomorphism of based rings, determining the image of each basis element $\mathsf{t}_w$ of the asymptotic algebra. As a consequence we verify that Lusztig's original conjecture from 1989 on the structure of the asymptotic algebra holds for this cell. A key ingredient in our proof is a refinement of Shi's description of~$Δ_1$, giving an explicit parametrisation of the elements of this cell (a \textit{cell factorisation}). This cell factorisation simultaneously describes all intersections of left and right cells, with uniqueness and length-additivity of expressions. Applications include a classification of the involutions in $Δ_1$, and a calculation of the length generating functions for $Δ_1$ and the right cells in~$Δ_1$.