On the Kazhdan--Lusztig order on cells and families

We consider the set $\text{Irr}(W)$ of (complex) irreducible characters of a finite Coxeter group $W$. The Kazhdan–Lusztig theory of cells gives rise to a partition of $\text{Irr}(W)$ into “families” and to a natural partial order ${\le }_{\text{cLR}}$ on these families. Following an idea of Spaltenstein, we show that ${\le }_{\text{cLR}}$ can be characterised (and effectively computed) in terms of standard operations in the character ring of $W$. If, moreover, $W$ is the Weyl group of an algebraic group $G$, then ${\le }_{\text{cLR}}$ can be interpreted, via the Springer correspondence, in terms of the closure relation among the “special” unipotent classes of $G$.