Commentarii Mathematici Helvetici

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Volume 87, Issue 4, 2012, pp. 905–927
DOI: 10.4171/CMH/273

Published online: 2012-10-10

On the Kazhdan–Lusztig order on cells and families

Meinolf Geck[1]

(1) Universität Stuttgart, Germany

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.

Keywords: Coxeter groups, Kazhdan–Lusztig cells, Springer correspondence

Geck Meinolf: On the Kazhdan–Lusztig order on cells and families. Comment. Math. Helv. 87 (2012), 905-927. doi: 10.4171/CMH/273