On generalized cells in affine Weyl groups
The concept of cells for an arbitrary Coxeter system \((W,S)\) was introduced by \textit{D. Kazhdan} and \textit{G. Lusztig} [in Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]. They define left, right, and two-sided cells, which play an important role in the study of the representations of the corresponding Hecke algebra. \textit{G. Lusztig} [Lect. Notes Math. 1024, 99-111 (1983; Zbl 0537.20019)], extends the concept of cells to the case in which there are integers \(C_s\geq 1\) associated to the simple reflections \(s\in S\) satisfying \(C_s=C_t\) if \(s,t\in S\) are conjugate. Integers subject to these conditions are referred to as parameters and the cells as generalized cells. The original cells are obtained if all \(C_s\) are equal. For equal parameters, cells in affine Weyl groups have been intensively studied. They have been explicitly described for type \(\widetilde A_n\), \(n\in\mathbb{N}\), rank 2, 3 and 4. Much less is known for unequal parameters. In this paper, the author determines the lowest generalized two-sided cell for affine Weyl groups. The author shows that it consists of at most \(|W_0|\) generalized left cells, where \(W_0\) denotes the corresponding finite Weyl group. For parameters coming from graph automorphisms, the author proves that this bound is exact. For such parameters, the author also characterizes all generalized left cells for finite and affine Weyl groups.
scientific article; zbMATH DE number 1019689
scientific article