Abstract
This chapter explains how the multiplication by \(T_{w_{0}}\) (where \({w_{0}}\) is the longest element of a finite Coxeter group) acts on Kazhdan–Lusztig basis elements: in particular, we show that it induces a non-trivial left (or right) cellular map, even if \({w_{0}}\) is central.
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Notes
Theorem 17.1.5 is due to Mathas in the split case [Mat] and to Lusztig [Lus21] in the general case. A simplification of Lusztig’s proof was provided by the author in [Bon5]: it avoids the use of the asymptotic algebra (which is defined in Chapter 19) and the use of (P15), which is sometimes the most difficult conjecture to prove. If W is a Weyl group, Losev [Los] proves that left multiplication by \(T_{w_0}\) may be viewed as an equivalence of categories involving the category \({\mathscr {O}}\).
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Bonnafé, C. (2017). Multiplication by \(T_{w_0}\). In: Kazhdan-Lusztig Cells with Unequal Parameters. Algebra and Applications, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-319-70736-5_17
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DOI: https://doi.org/10.1007/978-3-319-70736-5_17
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