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The inertial Jacquet–Langlands correspondence

Andrea Dotto

Abstract

We give a parametrization of the simple Bernstein components of inner forms of a general linear group over a local field by two invariants constructed from type theory, and explicitly describe its behaviour under the Jacquet–Langlands correspondence. Along the way, we prove a conjecture of Broussous, Sécherre and Stevens on preservation of endo-classes.

Funding source: Engineering and Physical Sciences Research Council

Award Identifier / Grant number: EP/L015234/1

Funding statement: This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London, and Imperial College London.

Acknowledgements

I thank Toby Gee for suggesting the problem which led to my involvement with this subject, and Colin Bushnell, Guy Henniart, Vincent Sécherre and Shaun Stevens for their advice and their interest in this work. The debt this paper owes to their ideas will be apparent to the reader, but this is a good place to acknowledge it explicitly. Finally, thanks are due to the referee for several useful comments.

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Received: 2021-04-15
Revised: 2021-09-09
Published Online: 2022-01-23
Published in Print: 2022-03-01

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