Balanced representations, the asymptotic Plancherel formula, and Lusztig’s conjectures for C 2
Algebraic Combinatorics, Volume 2 (2019) no. 5, p. 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.

Received : 2018-03-28
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 = {https://alco.centre-mersenne.org/item/ALCO_2019__2_5_969_0}
}
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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