JournalsjemsVol. 23, No. 9pp. 3129–3191

Galois self-dual cuspidal types and Asai local factors

  • U. K. Anandavardhanan

    Indian Institute of Technology Bombay, Mumbai, India
  • R. Kurinczuk

    Imperial College London, UK
  • Nadir Matringe

    Université de Poitiers, France
  • Vincent Secherre

    Université de Versailles, France
  • Shaun Stevens

    University of East Anglia, Norwich, UK
Galois self-dual cuspidal types and Asai local factors cover
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Let F/Fo\mathrm{F}/\mathrm{F}_{\mathsf{o}} be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and σ\sigma be its non-trivial automorphism. We show that any σ\sigma-self-dual cuspidal representation of GLn(F)\operatorname{GL}_n(\mathrm{F}) contains a σ\sigma-self-dual Bushnell–Kutzko type. Using such a type, we construct an explicit test vector for Flicker's local Asai L-function of a GLn(Fo)\operatorname{GL}_n(\mathrm{F}_\mathsf{o})-distinguished cuspidal representation and compute the associated Asai root number. Finally, by using global methods, we compare this root number to Langlands–Shahidi's local Asai root number, and more generally we compare the corresponding epsilon factors for any cuspidal representation.

Cite this article

U. K. Anandavardhanan, R. Kurinczuk, Nadir Matringe, Vincent Secherre, Shaun Stevens, Galois self-dual cuspidal types and Asai local factors. J. Eur. Math. Soc. 23 (2021), no. 9, pp. 3129–3191

DOI 10.4171/JEMS/1080