Conjugacy classes of involutions and Kazhdan-Lusztig cells. (Q2922913)

Let \((W,S)\) be a Coxeter system and let \(\varphi\) be a weight function of \(W\) in the sense of \textit{G. Lusztig} [see [1] = Hecke algebras with unequal parameters. Providence: AMS (2003; Zbl 1051.20003)]. For any \(X\subseteq W\), denote by \(\mathcal C(X)\) the union of all conjugacy classes of involutions in \(W\) which have non-empty intersection with \(X\). A two-sided cell \(\Gamma\) of \(W\) is called smooth if the family of irreducible characters associated with \(\Gamma\) contains only one element [see \textit{G. Lusztig}, Characters of reductive groups over a finite field. Princeton: Princeton University Press (1984; Zbl 0556.20033)].NEWLINENEWLINE Now let \(W\) be finite. Let \(\Gamma\) be a two-sided cell of \(W\) and let \(L\) and \(L'\) be two left cells of \(W\) contained in \(\Gamma\) [see [1], loc. cit.]. Then the main results of the present paper are as follows:NEWLINENEWLINE (a) If \(\varphi\) is constant on \(S\) then \(\mathcal C(L)=\mathcal C(L')\).NEWLINENEWLINE (b) Assume that Lusztig's Conjectures P1-P15 [in [1], loc. cit.] for \((W,S,\varphi)\) hold. If \(\Gamma\) is a smooth two-sided cell, then all the involutions in \(\Gamma\) are conjugate in \(W\). In particular, \(\mathcal C(L)=\mathcal C(L')\) is a single conjugacy class.NEWLINENEWLINE (c) Assume that \(\varphi\) is constant on \(S\) and that \(W\) is of type \(B_n\) or \(D_n\). Then for any conjugacy class \(\mathcal C\) of involutions in \(W\), the cardinals of the sets \(\mathcal C\cap L\) and \(\mathcal C\cap L'\) are the same.NEWLINENEWLINE When \(W\) is the symmetric group \(S_n\), \textit{M.-P. Schützenberger} showed that all the involutions contained in the same two-sided cell are conjugate [see Lect. Notes Math. 579, 59-113 (1977; Zbl 0398.05011)]. The present paper generalizes Schützenberger's result since all the two-sided cells of \(S_n\) are smooth.