Balanced representations, the asymptotic Plancherel formula, and Lusztig's conjectures for ${\tilde{C}}_2$

We prove Lusztig’s conjectures $\mathbf{P}\mathbf{1}$-$\mathbf{P}\mathbf{15}$ for the affine Weyl group of type ${\tilde{C}}_2$ 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 ${\bf P1}$-${\bf P15}$ follow. Next we establish an asymptotic Plancherel Theorem’ for type ${\tilde{C}}_2$, 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.