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Geometrizing the minimal representations of even orthogonal groups

Abstract : Let X be a smooth projective curve. Write Bun_{SO_{2n}} for the moduli stack of SO_{2n}-torsors on X. We give a geometric interpretation of the automorphic function f on Bun_{SO_{2n}} corresponding to the minimal representation. Namely, we construct a perverse sheaf K on Bun_{SO_{2n}} such that f should be equal to the trace of Frobenius of K plus some constant function. We also calculate K explicitely for curves of genus zero and one. The construction of K is based on some explicit geometric formulas for the Fourier coefficients of f on one hand, and on the geometric theta-lifting on the other hand. Our construction makes sense for more general simple algebraic groups, we formulate the corresponding conjectures. They could provide a geometric interpretation of some unipotent automorphic representations in the framework of the geometric Langlands program.
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https://hal.archives-ouvertes.fr/hal-01144678
Contributor : Vincent Lafforgue <>
Submitted on : Wednesday, April 22, 2015 - 2:01:06 PM
Last modification on : Wednesday, October 3, 2018 - 11:57:57 AM

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Vincent Lafforgue, Sergey Lysenko. Geometrizing the minimal representations of even orthogonal groups. Representation Theory, 2013, 17, pp.263-325. ⟨10.1090/S1088-4165-2013-00431-4⟩. ⟨hal-01144678⟩

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