Coxeter elements and Kazhdan-Lusztig cells
Let \((W,S,\Gamma)\) be a Coxeter system: a Coxeter group \(W\) with \(S\) the distinguished generator set and \(\Gamma\) the Coxeter graph. A Coxeter element of \(W\) is by definition a product of all generators \(s\in S\) in any fixed order. Let \(C(W)\) be the set of all the Coxeter elements in \(W\) and let \(C_0(W)=\bigcup_{J\subset S}C(W_J)\), where \(W_J\) is the standard parabolic subgroup of \(W\) generated by \(J\). For \(w\in C_0(W)\), denote by \(m(w)\) the maximal possible value of \(l(w_J)\) in an expression of the form \(w=x\cdot w_J\cdot y\). Denote by \(E(W)\) the set of all elements \(w\in C_0(W)\) such that \(m(sw)<m(w)\) for \(s\in{\mathcal L}(w)\), where \({\mathcal L}(w)=\{s\in S\mid l(sw)<l(w)\}\). In this paper the author gives an explicit expression for any distinguished involution \(d\) of \(W\) (\(W\) is of finite or affine type) with \(d\sim_Lw\) for some \(w\in E(W)\), i.e., if \(w=w_I\cdot y\) with \(m(w)=|I|\), then \(d=y^{-1}\cdot w_I\cdot y\), (for the notation \(d\sim_Lw\) and the definition of left cells see [\textit{D. Kazhdan} and \textit{G. Lusztig}, Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]). The result verifies a conjecture proposed by the author [Conjecture 8.10, in Adv. Sci. China, Math. 3, 79-98 (1990)]. A subset \(K\) of \(W\) is left-connected, if for any \(x,y\in K\), there exists a sequence of elements \(x=x_0,x_1,\dots,x_r=y\) in \(K\) such that \(x_{i-1}x_i^{-1}\in S\) for \(1\leq i\leq r\). The author also shows that any left cell of \(W\) containing some element of \(C_0(W)\) is left-connected, which verifies a conjecture of \textit{G. Lusztig} [Adv. Stud. Pure Math. 6 255-287 (1985; Zbl 0569.20032)] in this case.
scientific article; zbMATH DE number 1762097
scientific article