Balanced representations, the asymptotic Plancherel formula, and Lusztig’s conjectures for $\protect \tilde{C}_2$

Guilhot, Jérémie; Parkinson, James

doi:10.5802/alco.75

Balanced representations, the asymptotic Plancherel formula, and Lusztig’s conjectures for

{\tilde{C}}_{2}

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

Algebraic Combinatorics, Tome 2 (2019) no. 5, pp. 969-1031.

Résumé

We prove Lusztig’s conjectures P1–P15 for the affine Weyl group of type ${\tilde{C}}_{2}$ 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 ${\tilde{C}}_{2}$ , 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.

Reçu le : 2018-03-28
Révisé le : 2018-11-15
Accepté le : 2019-03-15
Publié le : 2019-10-08

DOI : 10.5802/alco.75

Classification : 16X00
Mots-clés : Kazhdan–Lusztig theory, Plancherel formula, affine Hecke algebras

Affiliations des auteurs :

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. https://www.numdam.org/articles/10.5802/alco.75/

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