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

We determine Lusztig’s asymptotic algebra for the second lowest two-sided cell~${\mathrm{\Delta }}_1$ in the equal parameter extended affine Weyl group ${\tilde{\mathsf{B}}}_n$ for all $n\ge 2$, realising it as a full matrix algebra over the representation ring of ${\mathsf{S}\mathsf{p}}_{2n-2}(\text{CC})\times C_2$ 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~${\mathrm{\Delta }}_1$, giving an explicit parametrisation of the elements of this cell (a $\mathit{\text{cell}}\mathit{\text{factorisation}}$). 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 ${\mathrm{\Delta }}_1$, and a calculation of the length generating functions for ${\mathrm{\Delta }}_1$ and the right cells in~${\mathrm{\Delta }}_1$.