Balanced representations, the asymptotic Plancherel formula, and Lusztig’s conjectures for C ˜ 2
Algebraic Combinatorics, Tome 2 (2019) no. 5, pp. 969-1031.

We prove Lusztig’s conjectures P1–P15 for the affine Weyl group of type C 2 for all choices of positive weight function. Our approach to computing Lusztig’s a-function is based on the notion of a “balanced system of cell representations”. Once this system is established roughly half of the conjectures P1–P15 follow. Next we establish an “asymptotic Plancherel Theorem” for type C 2 , from which the remaining conjectures follow. Combined with existing results in the literature this completes the proof of Lusztig’s conjectures for all rank 1 and 2 affine Weyl groups for all choices of parameters.

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DOI : 10.5802/alco.75
Classification : 16X00
Mots clés : Kazhdan–Lusztig theory, Plancherel formula, affine Hecke algebras
Guilhot, Jérémie 1 ; Parkinson, James 2

1 Institut Denis Poisson Université de Tours Université d’Orléans CNRS, Tours France
2 School of Mathematics and Statistics F07 University of Sydney NSW 2006, Australia
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Guilhot, Jérémie; Parkinson, James. Balanced representations, the asymptotic Plancherel formula, and Lusztig’s conjectures for $\protect \tilde{C}_2$. Algebraic Combinatorics, Tome 2 (2019) no. 5, pp. 969-1031. doi : 10.5802/alco.75. http://www.numdam.org/articles/10.5802/alco.75/

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