Complex Structure on the Smooth Dual of $GL(n)$

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 $\calh(\Omega)$ denote a Bernstein ideal in the Hecke algebra $\calh(G)$. 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 $\Pi^t(G)$.

2000 Mathematics Subject Classification: 46L80 22E50 46L87 11S37

Keywords and Phrases: Langlands correspondence, $p$-adic $GL(n)$, Baum-Connes map, smooth dual, tempered dual.

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