Celebratio Mathematica — Baum — Aubert/Plymen/Solleveld

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Works connected to Matilde Marcolli

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A.-M. Aubert, P. Baum, and R. Plymen: “The Hecke algebra of a reductive $p$ -adic group: A geometric conjecture,” pp. 1–34 in Noncommutative geometry and number theory: Where arithmetic meets geometry and physics (Bonn, Germany, August 2003 and June 2004). Edited by C. Consani and M. Marcolli. Aspects of Mathematics 37. Vieweg (Wiesbaden, Germany), 2006. MR 2327297 Zbl 1120.14001 incollection

Let $ \mathcal{H}(G) $ be the Hecke algebra of a reductive $ p $-adic group $ G $. We formulate a conjecture for the ideals in the Bernstein decomposition of $ \mathcal{H}(G) $. The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our conjecture is closely related to Lusztig’s conjecture on the asymptotic Hecke algebra. We prove our conjecture for $ \mathrm{SL}(2) $ and $ \mathrm{GL}(n) $. We also prove part (1) of the conjecture for the Iwahori ideals of the groups $ \mathrm{PGL}(n) $ and $ \mathrm{SO}(5) $. The conjecture, if true, leads to a parametrization of the smooth dual of $ G $ by the points in a complex affine locally algebraic variety.

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Caterina Consani	Related
Matilde Marcolli	Related