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Shuyang Cheng : Poisson summation for the Harish-Chandra transform

Classically the analytic properties of L-functions, in particular the functional equation, have been related to summation formulae of Poisson type. On the other hand, analytic properties of automorphic L-functions could be used to deduce functorial lifting of automorphic forms to general linear groups via the converse theorem. In his recent work, L. Lafforgue showed that a conjectural nonlinear Poisson summation formula on reductive groups is equivalent to the existence of functorial liftings to general linear groups. Here the Fourier transform is on a nonlinear space and involves nonstandard test functions. In my talk I will explain a toy model of such a summation formula for an integral transform between nonstandard spaces of test functions. The integral transform in question is the Harish-Chandra transform operating on the space of orbital integrals, and the summation formula follows from a trace formula on Lie algebras.

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