Kazhdan--Lusztig cells of $\mathbf{a}$-value 2 in $\mathbf{a}(2)$-finite Coxeter systems

Abstract

A Coxeter group is said to be \emph{$\mathbf{a}(2)$-finite} if it has finitely many elements of $\mathbf{a}$-value 2 in the sense of Lusztig. In this paper, we give explicit combinatorial descriptions of the left, right, and two-sided Kazhdan--Lusztig cells of $\mathbf{a}$-value 2 in an irreducible $\mathbf{a}(2)$-finite Coxeter group. In particular, we introduce elements we call \emph{stubs} to parameterize the one-sided cells and we characterize the one-sided cells via both star operations and weak Bruhat orders. We also compute the cardinalities of all the one-sided and two-sided cells.