Asymptotic Hecke Algebras and Lusztig–Vogan Bijection via Affine Matrix-Ball Construction

Abstract

Abstract The affine matrix-ball construction (abbreviated AMBC) was developed by Chmutov, Lewis, Pylyavskyy, and Yudovina as an affine generalization of the Robinson–Schensted correspondence. We show that AMBC gives a simple way to compute a distinguished involution in each Kazhdan–Lusztig cell of an affine symmetric group. We then use AMBC to give the 1st known canonical presentation for the asymptotic Hecke algebras of extended affine symmetric groups. As an application, we show that AMBC gives a conceptual way to compute the Lusztig–Vogan bijection. For the latter, we build upon prior works of Achar and Rush.