UvA-DARE (Digital Academic Repository) On cuspidal unipotent representations Feng, Y. Link to publication

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1 UvA-DARE (Digital Academic Repository) On cuspidal unipotent representations Feng, Y. Link to publication Citation for published version (APA): Feng, Y. (2015). On cuspidal unipotent representations General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam ( Download date: 18 Dec 2018

2 On Cuspidal Unipotent Representations Yongqi Feng

3 On Cuspidal Unipotent Representations ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. D.C. van den Boom ten overstaan van een door het College voor Promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op dinsdag, 8 september 2015, te 12:00 uur door Yongqi Feng geboren te Guangdong, China

4 Promotiecommissie: Promotor: prof. dr. E.M. Opdam (Universiteit van Amsterdam) Copromotor: dr. M.S. Solleveld (Radboud Universiteit Nijmegen) Overige leden: dr. R.R.J. Bocklandt (Universiteit van Amsterdam) dr. D.M. Ciubotaru (University of Oxford) prof. dr. V.J. Heiermann (Université d Aix-Marseille) dr. A.L. Kret (Universiteit van Amsterdam) prof. dr. S.V. Shadrin (Universiteit van Amsterdam) prof. dr. J.V. Stokman (Universiteit van Amsterdam) prof. dr. L.D.J. Taelman (Universiteit van Amsterdam) Faculteit der Natuurwetenschappen, Wiskunde en Informatica

5 On Cuspidal Unipotent Representations Yongqi Feng

6 To my grandparents

7 Contents 1 Introduction: Unipotent representations 1 2 Preliminaries Local factors Local class field theory The Weil group An example of split tori Root data Rational forms Pure inner forms Steinberg s vanishing theorm Unramified representations Spherical functions The Satake transform Macdonald s formula Matrix coe cients, formal degrees Lusztig s classification of unipotent representations Deligne-Lusztig characters Lusztig s series Cuspidal unipotent representations Unipotent representations of p-adic groups Hecke algebras as endomorphism algebras The Kazhdan-Lusztig parameters Formal degrees of cuspidal unipotent representations Local Langlands correspondence for unipotent representations Local Langlands parameters The Kottwitz isomorphism Enhanced discrete Langlands parameters The HII conjecture i

8 4 Spectral transfer maps of a ne Hecke algebras Generic a ne Hecke algebras Definitions The arithmetic diagrams The centre of H Casselman s criteria Spectral properties of a ne Hecke algebras Supports of Plancherel measures Definition of the µ-function Residual cosets Regularisation of µ along a residual coset Relations with Kazhdan-Lusztig parameters Spectral transfer maps Spectral transfer categories Examples of STM Spectral isomorphisms Spectal transfer morphisms of rank Spectral covering morphisms A uniqueness property Cuspidal formal degrees The spectral transfer category C class The essential uniqueness Classification of standard STMs Proof of the essential uniqueness Reduction to linear residual points The multiplicities of cyclotomic polynomials Reduction by using standard STMs The case = 1/ The case 2 {0, 1} Consequence of the essential uniqueness Check with classical groups Example: The group PCSp 2n An extra-special correspondence Description of the extra-special algorithm The extra-special spectral transfer morphisms The case of exceptional groups The split cases D 4 and 2 E Acknowledgements 128 ii

9 Samenvatting 129 Summary 131 Bibliography 133

10 According to the Doctorate Regulations of the University of Amsterdam, the following list and explanation is provided. The main body of this thesis is based on many intensive discussions between the author and his supervisor, as well as the following articles: (i) Y. Feng and E. M. Opdam, On a uniqueness property of cuspidal unipotent representations, arxiv: [math.rt] (ii) Y. Feng, Standard transfer morphisms for classical a appear. ne Hecke algebras. To The main result of this thesis is Theorem 5.4 in Section 5.5. The joint paper (i) deals with a big part of Theorem 5.4 (namely, the classical groups). The rest cases of Theorem 5.4 is treated in Chapter 7 (namely, the exceptional groups). The author proves the so-called extra special algorithm, which is one of the most important pieces of (i). Based on this algorithm he is able to verify the existence of the socalled extra special spectral transfer morphism (Chapter 6), which is necessary to prove the main result in (i). However, this job would never have been done without the patience, encouragement and support of the author s advisor. iv