Complex structure on the smooth dual of \(GL(n)\)
Let \(G= GL(n)\) be the general linear group over a \(p\)-adic field. The set of equivalence classes of irreducible smooth representations of \(G\) is called the smooth dual of \(G\) and denoted by \(\Pi(G)\). The local Langlands correspondence for \(GL(n)\) is a bijection between \(\Pi(G)\) and the set of Langlands parameters \(\Phi(G)\). The authors introduced in [\textit{J. Brodzki} and \textit{R. Plymen}, C. R. Acad. Sci. Paris, Sér. A 331, 213-218 (2000; Zbl 0970.46002)] a complex structure on the set of Langlands parameters of \(G\). This gives, via the local Langlands correspondence, a complex structure on the smooth dual of \(G\). In the paper under review, the authors describe in detail the complex structure on \(\Phi(G)\). Let \(\Pi(\Omega)\) denote a Bernstein component of \(\Pi(G)\) and let \(\mathcal{H}(\Omega)\) denote a Bernstein ideal in the Hecke algebra \(\mathcal{H}(G)\). The authors prove that the periodic cyclic homology of \(\mathcal{H}(\Omega)\) is isomorphic to the de Rham cohomology of \(\Pi(\Omega)\). They 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 \(\Pi^t(G)\).
scientific article; zbMATH DE number 1748375