On the Kazhdan–Lusztig order on cells and families

Abstract

We consider the set Irr( W ) of (complex) irreducible characters of a finite Coxeter group W . The Kazhdan–Lusztig theory of cells gives rise to a partition of Irr( W ) into “families” and to a natural partial order \leq_{\mathcal{LR}} on these families. Following an idea of Spaltenstein, we show that \leq_{\mathcal{LR}} can be characterised (and effectively computed) in terms of standard operations in the character ring of W . If, moreover, W is the Weyl group of an algebraic group G , then \leq_{\mathcal{LR}} can be interpreted, via the Springer correspondence, in terms of the closure relation among the “special” unipotent classes of G .