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On a local conjecture of Jacquet, ladder representations and standard modules

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Abstract

Let E / F be a quadratic extension of p-adic fields. We prove that every smooth irreducible ladder representation of the group \(GL_n(E)\) which is contragredient to its own Galois conjugate, possesses the expected distinction properties relative to the subgroup \(GL_n(F)\). This affirms a conjecture attributed to Jacquet for a large class of representations. Along the way, we prove a reformulation of the conjecture which concerns standard modules in place of irreducible representations.

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Notes

  1. Sometimes an additional assumption is added which requires the central character of \(\pi \) to be trivial on \(F^\times \). Note, that the counter-example below remains valid with this assumption.

  2. The proof in [4] adapts the results of [11] to its own setup. In particular, their \(W_2[M]\) is defined differently. Yet, going through the same proof verbatim with our definitions would give the same result for our setting.

  3. We will use this terminology to refer to a finite tuple of objects whose order is immaterial.

  4. We treat \(G_0 = P_1\) as the trivial group. We will formally refer to the one-dimensional irreducible representation of the trivial group 1, \(\varPsi ^+(1)\) and \(\varSigma (1)\) as empty representations in our notation. The product operation on them will have a trivial meaning \((1\times \pi = \pi )\).

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Acknowledgments

The work reported here is part of my Ph.D. research. I would like to thank my advisor, Omer Offen, for suggesting the problem, sharing his insights and providing much guidance. Thanks are also due to Erez Lapid, Nadir Matringe and Yiannis Sakellaridis who all supplied useful remarks during the work on this project.

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  1. Department of Mathematics, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Maxim Gurevich

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  1. Maxim Gurevich
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Correspondence to Maxim Gurevich.

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Gurevich, M. On a local conjecture of Jacquet, ladder representations and standard modules. Math. Z. 281, 1111–1127 (2015). https://doi.org/10.1007/s00209-015-1522-8

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