We prove Lusztig’s conjectures P1–P15 for the affine Weyl group of type for all choices of positive weight function. Our approach to computing Lusztig’s -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 , from which the remaining conjectures follow. Combined with existing results in the literature this completes the proof of Lusztig’s conjectures for all rank and affine Weyl groups for all choices of parameters.
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Keywords: Kazhdan–Lusztig theory, Plancherel formula, affine Hecke algebras
Guilhot, Jérémie 1; Parkinson, James 2
CC-BY 4.0
@article{ALCO_2019__2_5_969_0,
author = {Guilhot, J\'er\'emie and Parkinson, James},
title = {Balanced representations, the asymptotic {Plancherel} formula, and {Lusztig{\textquoteright}s} conjectures for $\protect \tilde{C}_2$},
journal = {Algebraic Combinatorics},
pages = {969--1031},
publisher = {MathOA foundation},
volume = {2},
number = {5},
year = {2019},
doi = {10.5802/alco.75},
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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, Volume 2 (2019) no. 5, pp. 969-1031. doi : 10.5802/alco.75. https://alco.centre-mersenne.org/articles/10.5802/alco.75/
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