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Opérateur de Lefschetz sur les tours de Drinfeld et Lubin–Tate

Part of: Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  25 January 2012

Jean-François Dat
Jean-François Dat*
Affiliation:
Université Pierre et Marie Curie, 4, place Jussieu, 75005, Paris, France (email: [email protected])
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Abstract

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We define and study a Lefschetz operator on the equivariant cohomology complex of the Drinfeld and Lubin–Tate towers. For ℓ-adic coefficients we show how this operator induces a geometric realization of the Langlands correspondence composed with the Zelevinski involution for elliptic representations. Combined with our previous study of the monodromy operator, this suggests a possible extension of Arthur’s philosophy for unitary representations occurring in the intersection cohomology of Shimura varieties to the possibly non-unitary representations occurring in the cohomology of Rapoport–Zink spaces. However, our motivation for studying the Lefschetz operator comes from the hope that its geometric nature will enable us to realize the mod-ℓ Langlands correspondence due to Vignéras. We discuss this problem and propose a conjecture.

Keywords

Lubin–Tate spaceslocal Langlands correspondence

MSC classification

Secondary: 14G35: Modular and Shimura varieties 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 2 , March 2012 , pp. 507 - 530
Copyright
Copyright © Foundation Compositio Mathematica 2012

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