Proof of the $p$-adic Kazhdan-Lusztig hypothesis for $\mathrm{GL}(n)$

Abstract

In this article, we prove the $p$-adic Kazhdan-Lusztig hypothesis for $\mathrm{GL}_n(F)$. While the approach via graded affine Hecke algebras due to recent work of Solleveld leads to more general results, this article serves to completes and clarifies the approach via affine Hecke algebras of Chriss and Ginzburg. In particular, this article serves as an opportunity to articulate several results which are undoubtedly known to experts, but have not been formally recorded in the literature.