Order relations on conjugacy classes and the Kazhdan-Lusztig map
A partition \(\lambda\) of \(n\in \mathbb{N}\) is a decreasing sequence of integers \(\lambda_ 1\geq\lambda_ 2\geq \dots\) such that \(\sum_{i\geq 1}\lambda_ i=n\), and write \(| \lambda|=n\). The set of all partitions of \(n\) is denoted \({\mathcal P}_ n\). For each \(i\), let \((\sum \lambda)_ i=\sum_{j\leq i}\lambda_ j\). There is an order relation on \({\mathcal P}=\bigcup_{n\in\mathbb{N}}{\mathcal P}_ n\) defined as follows. If \(\lambda,\mu\in{\mathcal P}\), define \(\lambda\leq \mu\) if \((\sum\lambda)_ i\leq (\sum\mu)_ i\) for every \(i\). If \({\mathfrak g}\cong {\mathfrak sl}_ n\), then the conjugacy classes in the Weyl group \(W\) are in bijective correspondence with \({\mathcal P}_ n\), and the conjugacy class in \(W\) corresponding to \(\lambda\in {\mathcal P}_ n\) is denoted \(C_ \lambda\). In this paper, the following theorem is shown: Let \({\mathfrak g}\cong{\mathfrak sl}_ n\) and \(\lambda,\mu\in{\mathcal P}_ n\). Then \(C_ \lambda\leq C_ \mu\) if and only if \(\lambda\leq \mu\). Let \({\mathcal X}={\mathcal P}\otimes{\mathcal P}\) and for each \(n\in\mathbb{N}\), let \({\mathcal X}_ n=\{x\in {\mathcal X}\mid | x|=n\}\). Similarly, if \({\mathfrak g}\cong{\mathfrak sp}_{2n}\), or \({\mathfrak so}_{2n+1}\) then the conjugacy classes in the Weyl group \(W\) are in bijective correspondence with \({\mathcal X}_ n\), and a result, which is analogous to the result mentioned above, for \({\mathfrak g}\cong {\mathfrak so}_{2n}\) is given in this paper, too. Moreover, a combinatorial description of the order relation on the set of conjugacy classes in the Weyl group \(W\) for classical Lie algebras is given. In particular, a map, defined in [\textit{D. Kazhdan, G. Lusztig} and \textit{J. Bernstein}, Isr. J. Math. 62, 129-168 (1988; Zbl 0658.22005)], which associates conjugacy classes in \(W\) to nilpotent orbits in \(\mathfrak g\) is studied.
scientific article; zbMATH DE number 56283