The asymptotic Plancherel formula and Lusztig's asymptotic algebra for An
Résumé
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.
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