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 \({\bf P1}\)–\({\bf P15}\) 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.

Keywords: Lusztig's conjectures, affine Hecke algebra.

AMS Subject Classification: Primary 20C08.