On the Kazhdan--Lusztig order on cells and families

Details On the Kazhdan--Lusztig order on cells and families On the Kazhdan--Lusztig order on cells and families

2009-10-24

arxiv.org/abs/0910.4689v3

Mathematics

Representation Theory

21 pages; to appear in Comm. Math. Helv. (2011); added reference in last revision

Scientific paper

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 $\leq_{\cLR}$ on these families. Following an idea of Spaltenstein, we show that $\leq_{\cLR}$ 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 $\leq_{\cLR}$ can be interpreted, via the Springer correspondence, in terms of the closure relation among the "special" unipotent classes of $G$.

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