Left cells in Weyl groups. (English) Zbl 0537.20019
Lie group representations I, Proc. Spec. Year, Univ. Md., College Park 1982-83, Lect. Notes Math. 1024, 99-111 (1983).
[For the entire collection see Zbl 0511.00011.]
In [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)] D. Kazhdan and G. Lusztig have defined a partition of an arbitrary Coxeter group into subsets called left cells. In the present paper, this is generalized to include the case where the simple reflections are given different weights. As a result it is shown that (12.1) when (W,S) is of type \(B_ n\) or \(D_ n\), any left cell in W carries a representation of W which is multiplicity free and has a number of irreducible components equal to a power of 2. Moreover the set of irreducible components can be organized in a natural way as a vector space over the field with 2 elements.
In [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)] D. Kazhdan and G. Lusztig have defined a partition of an arbitrary Coxeter group into subsets called left cells. In the present paper, this is generalized to include the case where the simple reflections are given different weights. As a result it is shown that (12.1) when (W,S) is of type \(B_ n\) or \(D_ n\), any left cell in W carries a representation of W which is multiplicity free and has a number of irreducible components equal to a power of 2. Moreover the set of irreducible components can be organized in a natural way as a vector space over the field with 2 elements.
Reviewer: Y.Asoo
MSC:
| 20G05 | Representation theory for linear algebraic groups |
