# ALGEBRAIC COMBINATORICS

Balanced representations, the asymptotic Plancherel formula, and Lusztig’s conjectures for ${\stackrel{\sim }{\mathit{C}}}_{2}$
Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 969-1031.

We prove Lusztig’s conjectures P1–P15 for the affine Weyl group of type ${\stackrel{\sim }{\mathit{C}}}_{2}$ for all choices of positive weight function. Our approach to computing Lusztig’s $\mathbf{a}$-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 ${\stackrel{\sim }{\mathit{C}}}_{2}$, 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.

Revised: 2018-11-15
Accepted: 2019-03-15
Published online: 2019-10-08
DOI: https://doi.org/10.5802/alco.75
Classification: 16X00
Keywords: Kazhdan–Lusztig theory, Plancherel formula, affine Hecke algebras
@article{ALCO_2019__2_5_969_0,
author = {Guilhot, J\'er\'emie and Parkinson, James},
title = {Balanced representations, the asymptotic Plancherel formula, and Lusztig's conjectures for ${\mathop {\protect \mathit{C}}\limits ^{\sim }}\_2$},
journal = {Algebraic Combinatorics},
publisher = {MathOA foundation},
volume = {2},
number = {5},
year = {2019},
pages = {969-1031},
doi = {10.5802/alco.75},
language = {en},
url = {alco.centre-mersenne.org/item/ALCO_2019__2_5_969_0/}
}
Guilhot, Jérémie; Parkinson, James. Balanced representations, the asymptotic Plancherel formula, and Lusztig’s conjectures for ${\mathop {\protect \mathit{C}}\limits ^{\sim }}_2$. Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 969-1031. doi : 10.5802/alco.75. https://alco.centre-mersenne.org/item/ALCO_2019__2_5_969_0/

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