The aim of this paper is to give a new explicit construction of Lusztig's asymptotic algebra in affine type 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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Chapelier‐Laget et al. (2024) studied this question.
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