The Hecke algebra of a reductive p-adic group: a geometric conjecture

Aubert, Anne-Marie, Baum, Paul and Plymen, Roger (2006) The Hecke algebra of a reductive p-adic group: a geometric conjecture. In, Consani, C. and Marcolli, M. (eds.) Noncommutative Geometry and Number Theory: Where Arithmetic meets Geometry and Physics. (Vieweg Aspects of Mathematics, 37) Wiesbaden, DE. Vieweg, pp. 1-34. (doi:10.1007/978-3-8348-0352-8_1).

Abstract

Let 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 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 SL(2) and GL(n). We also prove part (1) of the conjecture for the Iwahori ideals of the groups PGL(n) and 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.

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