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

J. Guilhot and J. Parkinson

Abstract

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 $\mathbf{P}\mathbf{1}$–$\mathbf{P}\mathbf{15}$ 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.

Keywords: Lusztig's conjectures, affine Hecke algebra.

AMS Subject Classification: Primary 20C08.