Endo-parameters for p -adic classical groups
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Endo-parameters for p-adic classical groups Robert Kurinczuk1 · Daniel Skodlerack2,3 · Shaun Stevens3
Received: 9 January 2017 / Accepted: 20 August 2020 © The Author(s) 2020
Abstract For a classical group over a non-archimedean local field of odd residual characteristic p, we prove that two cuspidal types, defined over an algebraically closed field C of characteristic different from p, intertwine if and only if they are conjugate. This completes work of the first and third authors who showed that every irreducible cuspidal C-representation of a classical group is compactly induced from a cuspidal type. We generalize Bushnell and Henniart’s notion of endo-equivalence to semisimple characters of general linear groups and to self-dual semisimple characters of classical groups, and introduce (self-dual) endo-parameters. We prove that these parametrize intertwining classes of (self-dual) semisimple characters and conjecture that they are in bijection with wild Langlands parameters, compatibly with the local Langlands correspondence.
B Daniel Skodlerack
[email protected] Robert Kurinczuk [email protected] Shaun Stevens [email protected]
1
Department of Mathematics, Imperial College, London SW7 2AZ, UK
2
Institute of Mathematical Sciences, ShanghaiTech University, Pudong 201210, China
3
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK
123
R. Kurinczuk et al.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Witt groups and transfer . . . . . . . . . . . . . . . . . . . . . 4 Classical groups . . . . . . . . . . . . . . . . . . . . . . . . . 5 Simple strata and concordance . . . . . . . . . . . . . . . . . . 6 Self-dual simple characters: intertwining and concordance . . . 7 Self-dual ps-characters and simple endo-classes . . . . . . . . 8 Self-dual semisimple characters: intertwining and concordance 9 Self-dual semisimple endo-classes . . . . . . . . . . . . . . . . 10 Intertwining and conjugacy for special orthogonal groups . . . 11 Intertwining implies conjugacy for cuspidal types . . . . . . . 12 Endo-parameters . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: From skew to self-dual semisimple characters . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction 1.1. One approach to study smooth representations of a reductive p-adic group on modules over a commutative ring intrinsically is by restriction to compact open subgroups. For p-adic general linear groups this has yielded detailed classification results, for example Bushn
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