We prove Lusztig's conjectures P1–P15 for the affine Weyl group of type G∼2 for all choices of parameters. Our approach to compute Lusztig's bolda‐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 bolda‐function and we explicitly construct such a system in type G∼2 for arbitrary parameters. We then investigate the connection between Kazhdan–Lusztig cells and the Plancherel theorem in type G∼2, allowing us to prove 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 G2 and A1, along with some explicit computations for the finite cells.
Abstract. Bremke and Xi determined the lowest two-sided cell for affine Weyl groups with unequal parameters and showed that it consists of at most |W 0 | left cells where W 0 is the associated finite Weyl group. We prove that this bound is exact. Previously, this was known in the equal parameter case and when the parameters were coming from a graph automorphism. Our argument uniformly works for any choice of parameters.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.