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 \({\bf P1}\)–\({\bf P15}\) 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 \({\bf P1}\) 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.