The partial order on two-sided cells of certain affine Weyl groups
In their famous paper [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)], \textit{D. Kazhdan} and \textit{G. Lusztig} introduced the concept of equivalence classes such as left cell, right cell and two-sided cell in a Coxeter group \(W\). Concerning an affine Weyl group \(W_a\), \textit{G. Lusztig} showed that the set \(\text{Cell}(W_a)\) of two-sided cells of \(W_a\) is in a natural 1-1 correspondence with the set \({\mathfrak U}(G)\) of unipotent classes in the corresponding algebraic group [J. Fac. Sci. Univ. Tokyo, Sect. I A 36, No. 2, 297-328 (1989; Zbl 0688.20020)]. In his conjecture \(D\) [of ``Algebraic Groups and Related Topics'', Adv. Stud. Pure Math. 6, 255-287 (1985; Zbl 0569.20032)], \textit{G. Lusztig} conjectured that there exists an order-preserving bijection between the set \(\text{Cell}(W_a)\) of two-sided cells of \(W_a\) and the set \({\mathfrak U}(G)\) of unipotent conjugacy classes of the corresponding algebraic group \(G\). In this paper, the author gives an affirmative answer of the conjecture in the cases when \(W_a\) is of type \(\widetilde{A}_{n-1}\) and when \(W_a\) has rank 4.
scientific article; zbMATH DE number 843882
scientific article