The asymptotic Plancherel formula and Lusztig's asymptotic algebra for $\tilde{\mathsf{A}}_n$

TL;DR

This paper introduces a new explicit construction of Lusztig's asymptotic algebra for affine type A, utilizing a balanced system of cell modules, an asymptotic Plancherel theorem, and a relative Satake Isomorphism.

Contribution

It provides the first explicit construction of Lusztig's asymptotic algebra in affine type A with new methods involving cell modules and asymptotic harmonic analysis.

Findings

Constructed a balanced system of cell modules.

Proved an asymptotic version of the Plancherel Theorem.

Developed a relative Satake Isomorphism for two-sided Kazhdan-Lusztig cells.

Abstract

The aim of this paper is to give a new explicit construction of Lusztig's asymptotic algebra in affine type $\mathsf{A}$. To do so, we construct a balanced system of cell modules, prove an asymptotic version of the Plancherel Theorem and develop a relative version of the Satake Isomorphism for each two-sided Kazhdan-Lusztig cell.