On the Kazhdan-Lusztig order on cells and families.
Let \(\mathrm{Irr}(W)\) be the set of irreducible characters of a finite Coxeter group \(W\) over the complex number field \(\mathbb C\). Then \(\mathrm{Irr}(W)\) can be partitioned in two ways: one is in terms of Kazhdan-Lusztig two-sided cells of \(W\) (see \textit{D. Kazhdan} and \textit{G. Lusztig}, [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]), another is in terms of families [\textit{G. Lusztig}, Characters of reductive groups over a finite field. Ann. Math. Stud. 107. Princeton: Princeton University Press (1984; Zbl 0556.20033)]. These two partitions are the same. The theory of Kazhdan-Lusztig cells provides a natural partial order \(\leqslant_{LR}\) on the pieces of the partition. The main purpose of the present paper is to obtain a better understanding of the partial order \(\leqslant_{LR}\). The author shows that \(\leqslant_{LR}\) can be characterized in a purely elementary way in terms of standard operations (i.e., induction, truncated induction, tensoring with sign) in the character ring of \(W\). An efficient algorithm for computing the partial order can be implemented in CHEVIE, [\textit{M. Geck} et al., Appl. Algebra Eng. Commun. Comput. 7, No. 3, 175-210 (1996; Zbl 0847.20006)]. When \(W\) is the Weyl group of an algebraic group \(G\), the author shows that the partial order \(\leqslant_{LR}\) on the families of \(\mathrm{Irr}(W)\) can be interpreted via the Springer correspondence in terms of the closure relation among the special unipotent classes of \(G\). Note that a version of the latter result appeared in the paper by \textit{D. Barbasch} and \textit{D. A. Vogan}, [Ann. Math. (2) 121, 41-110 (1985; Zbl 0582.22007)]. The author's proof is based on the results of \textit{N. Spaltenstein}, [J. Reine Angew. Math. 343, 212-220 (1983; Zbl 0503.20012)], which is different from that of \textit{D. Barbasch} and \textit{D. A. Vogan} [loc. cit.].
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