A proof of Lusztig's conjectures for affine type $G_2$ with arbitrary parameters

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 ${\tilde{G}}_2$ for all choices of parameters. Our approach to compute Lusztig's $\mathbf{a}$-function is based on the notion of a "balanced system of cell representations" for the Hecke algebra. We show that for arbitrary Coxeter type the existence of balanced system of cell representations is sufficient to compute the $\mathbf{a}$-function and we explicitly construct such a system in type ${\tilde{G}}_2$ for arbitrary parameters. We then investigate the connection between Kazhdan-Lusztig cells and the Plancherel Theorem in type ${\tilde{G}}_2$, allowing us to prove $\mathbf{P}\mathbf{1}$ and determine the set of Duflo involutions. From there, the proof of the remaining conjectures follows very naturally, essentially from the combinatorics of Weyl characters of types $G_2$ and $A_1$, along with some explicit computations for the finite cells.

Keywords: Lusztig conjectures, Hecke algebra, Kazhdan-Lustig polynomial.

AMS Subject Classification: Primary 20C08; secondary 05E10.