Asymptotic lowest two-sided cell

To a Coxeter system $(W,S)$ (with $S$ finite) and a weight function $L : W \to \NM$isassociatedapartitionof$W$ into Kazhdan-Lusztig (left, right or two-sided) $L$-cells. Let $S^{\circ }=\{s\in S\mathrm{\mid }L(s)=0\}$, $S^+={s \in S | L(s) > 0}$andlet$C$beaKazhdan-Lusztig(left,rightortwo-sided)$L$-cell. According to the semicontinuity conjecture of the first author, there should exist a positive natural number $m$ such that, for any weight function $L^{\mathrm{\prime }}:W\to \text{NM}$ such that $L(s^{+})=L^{\mathrm{\prime }}(s^{+})>m{L}^{\mathrm{\prime }}(s^{\circ })$ for all $s^+ \in S^+$and$s^\circ \in S^\circ$,$C$ is a union of Kazhdan-Lusztig (left, right or two-sided) $L^{\mathrm{\prime }}$-cells. The aim of this paper is to prove this conjecture whenever $(W,S)$ is an affine Weyl group and $C$ is contained in the lowest two-sided $L$-cell.