Complex structure on the smooth dual of $GL(n)$

Abstract

Let $G$ denote the $p$-adic group $GL(n)$, let $\Pi (G)$ denote the smooth dual of $G$, let $\Pi (\Omega )$ denote a Bernstein component of $\Pi (G)$ and let $\mathcal{H}(\Omega )$ denote a Bernstein ideal in the Hecke algebra $\mathcal{H}(G)$. With the aid of Langlands parameters, we equip $\Pi (\Omega )$ with the structure of complex algebraic variety, and prove that the periodic cyclic homology of $\mathcal{H}(\Omega )$ is isomorphic to the de Rham cohomology of $\Pi (\Omega )$. We show how the structure of the variety $\Pi (\Omega )$ is related to Xi's affirmation of a conjecture of Lusztig for $GL(n,\mathbb{C})$. The smooth dual $\Pi (G)$ admits a deformation retraction onto the tempered dual ${\Pi }^t(G)$.