Geodesic
Complex structure on the smooth dual of $GL(n)$
Documenta mathematica, Tome 7 (2002), pp. 91-112.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $\Pi^t(G)$.
Classification : 46L80, 22E50, 46L87, 11S37
Keywords: Langlands correspondence, $p$-adic $GL(n)$, baum-connes map, smooth dual, tempered dual 
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     author = {Brodzki, Jacek and Plymen, Roger},
     title = {Complex structure on the smooth dual of $GL(n)$},
     journal = {Documenta mathematica},
     pages = {91--112},
     publisher = {mathdoc},
     volume = {7},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2002__7__a16/}
}
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Brodzki, Jacek; Plymen, Roger. Complex structure on the smooth dual of $GL(n)$. Documenta mathematica, Tome 7 (2002), pp. 91-112. http://geodesic.mathdoc.fr/item/DOCMA_2002__7__a16/