Balanced representations, the asymptotic Plancherel formula, and Lusztig’s conjectures for ${\mathop {\protect \mathit{C}}\limits ^{\sim }}_2$

Abstract

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.