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On positivity in Hecke algebras. (English) Zbl 0717.20027

Let \(\mathcal H\) be the Hecke algebra associated with a Coxeter system \((W,R)\) (i.e. \(W\) is a Coxeter group and \(R\) its set of distinguished involutory generators). D. Kazhdan and G. Lusztig [Invent. Math. 53, 165–184 (1979; Zbl 0499.20035)] introduced new bases of \(\mathcal H\), with respect to which the structure constants (which are Laurent polynomials) satisfy remarkable positivity properties. These properties were explained by interpreting the structure constants geometrically as Poincaré series. The authors of this paper point out some further positivity properties for the structure constants of \(\mathcal H\), prove some of them formally from existing results, others geometrically using intersection complexes on flag varieties, and state the unproved cases as conjectures.
Reviewer: Fuan Li (Beijing)

MSC:

20G05 Representation theory for linear algebraic groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
16G99 Representation theory of associative rings and algebras
16U99 Conditions on elements
14L30 Group actions on varieties or schemes (quotients)

Citations:

Zbl 0499.20035
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