Balanced representations, the asymptotic Plancherel formula, and Lusztig's conjectures for C2

Jérémie Guilhot (1) , James Parkinson (2)

1 IDP - Institut Denis Poisson
2 The University of Sydney

We prove Lusztig's conjectures P1–P15 for the affine Weyl group of type˜ C2 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 C2, 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.

Domaines: Théorie des représentations [math.RT]

Jérémie Guilhot, James Parkinson. Balanced representations, the asymptotic Plancherel formula, and Lusztig's conjectures for C2. Algebraic Combinatorics, 2019, 2 (5), pp.969-1031. ⟨10.5802/alco.75⟩. ⟨hal-01745431⟩