Classification of Coxeter groups with finitely many elements of \(\mathfrak{a}\)-value 2 (Q1985355)

This paper concerns the Kazhdan-Lusztig basis of a Coxeter group \(W\), and the associated \(\mathbf a\)-function \(W\to\mathbb N\). This function is defined in terms of the structure constants for the Kazhdan-Lusztig basis, and is important in understanding the Kazhdan-Lusztig cells of \(W\); in particular, the function is constant on each cell. The aim of the paper is a straightforward one: to determine which Coxeter groups are \textit{\(\mathbf a(2)\)-finite}, i.e. have only finitely many elements \(w\) with \(\mathbf a(w)=2\). This means that in some sense the Coxeter group is ``not too infinite'', and the main theorem gives a complete classification of \(\mathbf a(2)\)-finite groups in terms of Coxeter graphs. Interestingly, the classification includes some, but not all, Coxeter groups of affine type. The proof involves a variety of tools, mostly managing to avoid explicit work with the Kazhdan-Lusztig basis: some known results on \(\mathbf a\)-values (in particular, those relating to the ``star operations''), and study of heaps associated to fully commutative elements of Coxeter groups. The paper is very well written, giving an excellent introduction to the subject and the tools used, and presenting the proof with a suitable level of detail.