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.

: Primary 20C08.

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 Wednesday, March 28, 2018