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

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