Unequal-parameter left, right, and two-sided cells and cell quotients

Let $C_w$ be the basis of Existence and uniqueness of the ordered-group unequal-parameter Kazhdan–Lusztig basis. For $x,y\in W$, write $x{\leftarrow }_{L}y$ if the coefficient of $C_x$ in $C_s{C}_y$ is nonzero for some $s\in S$. Write $x{\le }_{L}y$ if there is a finite chain $x=w_0,\dots ,w_n=y$ with $w_i{\leftarrow }_L{w}_{i+1}$ at every step. Define $x{\leftarrow }_{R}y$ from the coefficient of $C_x$ in $C_y{C}_s$, and define ${\le }_R$ by finite chains of such steps. Define $x{\le }_{LR}y$ by finite chains in which each step is either a left or a right step. The length-zero chain makes each relation reflexive, and concatenation makes each transitive. Mutual comparability defines equivalence relations; their classes are the left, right, and two-sided cells.

For a left cell $\mathcal{C}$, define $I_{\le \mathcal{C}}^L={\mathrm{span}}_A\{C_x:x{\le }_{L}y\text{ for some }y\in \mathcal{C}\},I_{<\mathcal{C}}^L={\mathrm{span}}_A\{C_x:x{\le }_{L}y\text{ for some }y\in \mathcal{C},x\mathrm{\notin }\mathcal{C}\}\mathrm{.}$ Define $I_{\le \mathcal{C}}^R,I_{<\mathcal{C}}^R$ for a right cell and $I_{\le \mathcal{C}}^{LR},I_{<\mathcal{C}}^{LR}$ for a two-sided cell in the same way, using the corresponding preorder. The associated left cell quotient, right cell quotient, and two-sided cell quotient are respectively $I_{\le \mathcal{C}}^L\mathrm{/}{I}_{<\mathcal{C}}^L$, $I_{\le \mathcal{C}}^R\mathrm{/}{I}_{<\mathcal{C}}^R$, and $I_{\le \mathcal{C}}^{LR}\mathrm{/}{I}_{<\mathcal{C}}^{LR}$. Weighted cell quotients are well-defined modules and bimodules ↗ proves that these quotients inherit the stated module or bimodule actions and have bases indexed by their cells.