DOCSLIB.ORG

Aspects of Automorphic Induction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Aspects of Automorphic Induction Aspects of Automorphic Induction DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Edward Michael Belfanti, Jr., B.A. Graduate Program in Mathematics The Ohio State University 2018 Dissertation Committee: James W. Cogdell, Advisor Sachin Gautam Roman Holowinsky c Copyright by Edward Michael Belfanti, Jr. 2018 Abstract Langlands' functoriality conjectures predict how automorphic representations of different groups are related to one another. Automorphic induction is a basic case of functoriality motivated by Galois theory. Let F be a local field of characteristic 0. The local Langlands correspondence for GL(N) states that there is a bijection between N-dimensional, complex 0 representations of the Weil-Deligne group WF and irreducible, admissible representations of GL(N; F ). Given an operation on Weil group representations, one can ask what the corresponding operation is for representations of GL(N; F ). Automorphic induction is the operation on representations of GL(N; F ) corresponding to induction of representations of the Weil group. Given a cyclic extension E ⊃ F of degree D, automorphic induction is a mapping of representations of GL(M; E) to representations of GL(MD; F ). Once automorphic induction has been established for local fields, one can ask if the operation applied at each local component of an automorphic representation produces another automorphic representation. The automorphic induction problem was first considered in detail in [Kaz84]. Kazhdan proved the local automorphic induction map exists in the of M = 1. The next major result was [AC89], where it was shown that a global automorphic induction operation exists for prime degree extensions. The local theory was completed in [HH95]; Henniart and Herb showed that the local automorphic induction map exists for arbitrary M and cyclic extensions of arbitrary degree. Moreover, Henniart showed in [Hen01] that the local automorphic induction map constructed in [HH95] is consistent with induction of Weil group representations and the local Langlands correspondence. Finally, Henniart extended the results of [AC89] in [Hen12] to cyclic extensions of any degree and verified that the resulting ii mapping of automorphic representations is consistent with the local lifting of [HH95]. All of these results rely on some version of the trace formula and powerful theorems about L-functions for GL(N). The goal of this thesis is to give a different proof of local and global automorphic induction when M = 1 and to emphasize some different aspects of the theory. As with the results previously mentioned, the main technical tool is a trace formula, specifically the full global trace formula of Arthur. The proof relies more on the trace formula and less on L-functions than previous proofs. Chapter 1 consists of background and preliminaries needed to state the main theorems on local and global automorphic induction. We also discuss various results that are needed to state Arthur's trace formula. In Chapter 2, we give a statement of the relevant trace formulas. We also discuss some computations in general rank that could be used in a generalization of our techniques to the case of M > 1. The main local and global theorems are established in Chapter 3. As with most trace formula arguments, the proof boils down to an application of linear independence of characters. iii For my family, friends, and all those who helped along the way iv Acknowledgments A dissertation is the work of many people, some of whom deserve special thanks. Melissa Grasso has been with me through not one, but two final semesters during which I was convinced I would never graduate. Her love, support and companionship over the past seven years, and through some particularly trying months, has helped me more than I can ever repay. Her own successes and diligent hard work provided the motivation I needed to see this project through to the end; I only hope that the completion of this degree justifies in some small way the innumerable sacrifices, big and small, she has made for me. With love, thank you. My parents, Ed and Teresa, have provided boundless love, generosity and advice, not just these past seven years but the past twenty-nine. I feel fortunate every day for their support, and it is an honor to be their son and share any success I may find with them. Thank you always. My brothers, Nick and Dom, have provided the support and encouragement that their older brother is supposed to provide. Thank you both for never doubting me. Thanks to Nana, Papa, and Grandma, for the love and support unique to grandparents. Thanks to my Grandpa, who I know would be proud of me. Thanks to my aunts and uncles Amy, Paul and Mary, and my cousins William, Charles and Kate, for a fun Christmas before getting back to the grind. I've received many kinds words from my aunts Nancy, Mary, Donna and Julie, which were especially welcome over the last weekend of work on this document. I would like to thank Linda James, John Grasso, Jennifer Grasso, Erika James and Kevin Riopelle, for always making me feel welcome in Sudbury. v Stephen Eigenmann deserves thanks for always helping me keep my priorities straight. Thanks to Jack Marczak, with whom I started studying math in high school. Thanks also to Larry Marczak, without whose help I may never have mastered logarithms. I would like to thank the group chat consisting of Matt Flagg, Eric Kim, Rob Monaco and Tomas Castella for providing needed humor over the past seven years and for keeping me apprised of the news and scores I missed while completing this work. Matt deserves special thanks for his commiseration and support throughout more than ten years of school. John Lynch and I have been friends ever since we realized we were in so many classes together that it was getting weird for us not to talk. Carney forever. I would like to thank my friends and former roommates Steph Wilker, Scott Jaffee and Matt Jaffee, who managed to tolerate me through my early years in grad school, and qualifying exams in particular. Thanks to Catherine Glover and Lyman Gillispie for their friendship, and for getting me out of my office. A special thanks to Lyman, for always taking seriously my terrible opinions about books that I don't understand. I would like to thank my colleague Yilong Wang for interesting conversations, mathe- matical and otherwise, and for encouragement over the last few weeks of writing. Dan Moore, PhD, has been a close friend ever since he got over my gum snapping. His technical support kept me sane at a very strenuous time during the completion of this document. Thanks for the many, let's say bizarre and enlightening, conversations. We'll always have that Culver's on the outskirts of middle of nowhere Indiana. Dan could not have supported me without the support he received from Mariel Colman, who deserves special thanks for her patience with us. Thanks to Solomon Friedberg, whose advice and encouragement when I was at Boston College inspired me to go to graduate school, and study representation theory in particular. Jim Cogdell, my advisor, deserves thanks for many hours of conversation about repre- sentation theory, as well as for frantically reading this thesis and providing feedback as I pushed every conceivable deadline. It is impossible to believe this document could exist without his wisdom and advice throughout the research and writing process. He deserves vi special thanks for helping me through a difficult project and situation not of his design. vii Vita 2007{2011 . Boston College B.A. in Mathematics 2011{Present . The Ohio State University Graduate Teaching Associate, Graduate Research Associate Fields of Study Major Field: Mathematics Studies in Automorphic Representation Theory: James W. Cogdell viii Table of Contents Page Abstract........................................... ii Dedication......................................... iv Acknowledgments.....................................v Vita............................................. viii Chapters 1 Preliminaries ..................................... 1 1.1 L-groups...................................... 1 1.2 Twisted endoscopy for (GN ;!) ......................... 11 1.3 Representations of GN .............................. 22 1.4 A substitute for global parameters ....................... 44 1.5 Langlands-Shelstad-Kottwitz transfer...................... 64 1.6 Local intertwining operators........................... 79 1.7 Statement of the main theorems ........................ 99 2 Trace formulas .................................... 108 2.1 The discrete part of the trace formula ..................... 108 2.2 Stabilization.................................... 114 2.3 Contribution of a parameter . 121 2.4 A preliminary comparison............................ 128 2.5 The stable multiplicity formula......................... 135 2.6 An elliptic computation ............................. 146 2.7 Globalizing local representations ........................ 153 2.8 Orthogonality relations.............................. 164 3 The case of M = 1.................................. 170 3.1 Stable multiplicity when M = 1......................... 170 3.2 Orthogonality relations when M = 1...................... 171 3.3 Spherical characters and weak lifting...................... 173 3.4 Main theorems .................................. 178 Bibliography ....................................... 191 ix Appendices A Odds and ends ...................................
Load more

Recommended publications
  • Reducible Principal Series Representations, and Langlands Parameters for Real Groups3
    REDUCIBLE PRINCIPAL SERIES REPRESENTATIONS, AND LANGLANDS PARAMETERS FOR REAL GROUPS DIPENDRA PRASAD May 15, 2018 Abstract. The work of Bernstein-Zelevinsky and Zelevinsky gives a good under- standing of irreducible subquotients of a reducible principal series representation of GLn(F), F a p-adic field induced from an essentially discrete series representation of a parabolic subgroup (without specifying their multiplicities which is done by a Kazhdan-Lusztig type conjecture). In this paper we make a proposal of a similar kind for principal series representations of GLn(R). Our investigation on principal series representations naturally led us to consider the Steinberg representation for real groups, which has curiuosly not been paid much attention to in the subject (unlike the p-adic case). Our proposal for Steinberg is the simplest possible: for a real reductive group G, the Steinberg of G(R) is a discrete series representation if and only if G(R) has a discrete series, and makes up a full L-packet of representations of G(R) (of size WG/WK), so is typically not irreducible. Contents 1. Introduction 1 2. Notion of Segments for GLn(R) 4 3. The Steinberg representation for real groups 4 4. Paraphrasing the work of Speh 8 5. A partial order on the set of irreducible representations 9 6. Principal series representations for GL3(R), GL4(R) 9 7. An Example 10 References 11 arXiv:1705.01445v2 [math.RT] 14 May 2018 1. Introduction Although irreducible representations of GLn(R) and GLn(C) are well understood, and so are their Langlands parameters, the author has not found a place which dis- cusses Langlands parameters of irreducible subquotients of principal series represen- tations on these groups.
    [Show full text]
  • An Introduction to the Trace Formula
    Clay Mathematics Proceedings Volume 4, 2005 An Introduction to the Trace Formula James Arthur Contents Foreword 3 Part I. The Unrefined Trace Formula 7 1. The Selberg trace formula for compact quotient 7 2. Algebraic groups and adeles 11 3. Simple examples 15 4. Noncompact quotient and parabolic subgroups 20 5. Roots and weights 24 6. Statement and discussion of a theorem 29 7. Eisenstein series 31 8. On the proof of the theorem 37 9. Qualitative behaviour of J T (f) 46 10. The coarse geometric expansion 53 11. Weighted orbital integrals 56 12. Cuspidal automorphic data 64 13. A truncation operator 68 14. The coarse spectral expansion 74 15. Weighted characters 81 Part II. Refinements and Applications 89 16. The first problem of refinement 89 17. (G, M)-families 93 18. Localbehaviourofweightedorbitalintegrals 102 19. The fine geometric expansion 109 20. Application of a Paley-Wiener theorem 116 21. The fine spectral expansion 126 22. The problem of invariance 139 23. The invariant trace formula 145 24. AclosedformulaforthetracesofHeckeoperators 157 25. Inner forms of GL(n) 166 Supported in part by NSERC Discovery Grant A3483. c 2005 Clay Mathematics Institute 1 2 JAMES ARTHUR 26. Functoriality and base change for GL(n) 180 27. The problem of stability 192 28. Localspectraltransferandnormalization 204 29. The stable trace formula 216 30. Representationsofclassicalgroups 234 Afterword: beyond endoscopy 251 References 258 Foreword These notes are an attempt to provide an entry into a subject that has not been very accessible. The problems of exposition are twofold. It is important to present motivation and background for the kind of problems that the trace formula is designed to solve.
    [Show full text]
  • Reducibility of Generalized Principal Series Representations
    REDUCIBILITY OF GENERALIZED PRINCIPAL SERIES REPRESENTATIONS BY BIRGIT SPEH and DAVID A. VOGAN, JR(1) Mathematische Institut Massachusetts Institute der Universitlit Bonn of Technology, Cambridge Bonn, West Germany Massachusetts, U.S.A. 1. Introduction Let G be a connected semisimple matrix group, and P___ G a cuspidal parabolic~sub- group. Fix a Langlands decomposition P = MAN of P, with N the unipotent radical and A a vector group. Let ~ be a discrete series re- presentation of M, and v a (non-unitary) character of A. We call the induced representation ~(P, ~ | v) = Ind~ (6 | v | 1) (normalized induction) a generalized principal series representation. When v is unitary, these are the representations occurring in Harish-Chandra's Plancherel formula for G; and for general v they may be expected to play something of the same role in harmonic analysis on G as complex characters do in R n. Langlands has shown that any irreducible admissible representation of G can be realized canonically as a subquotient of a generalized principal series representation (Theorem 2.9 below). For these reasons and others (some of which will be discussed below) one would like to understand the reducibility of these representa- tions, and it is this question which motivates the results of this paper. We prove THEOREM 1.1. (Theorems 6.15 and 6.19). Let g(P, 5| be a generalized principal series representation. Fix a compact Caftan subgroup T + o/M (which exists because M has a discrete series). Let ~)= t++a, g (') Support by an AMS Research Fellowship. 228 B. SPEll AlgD D.
    [Show full text]
  • CUSPIDAL IRREDUCIBLE COMPLEX OR L-MODULAR REPRESENTATIONS of QUATERNIONIC FORMS of P-ADIC CLASSICAL GROUPS for ODD P
    CUSPIDAL IRREDUCIBLE COMPLEX OR l-MODULAR REPRESENTATIONS OF QUATERNIONIC FORMS OF p-ADIC CLASSICAL GROUPS FOR ODD p DANIEL SKODLERACK Abstract. Given a quaternionic form G of a p-adic classical group (p odd) we clas- sify all cuspidal irreducible representations of G with coefficients in an algebraically closed field of characteristic different from p. We prove two theorems: At first: Every irreducible cuspidal representation of G is induced from a cuspidal type, i.e. from a certain irreducible representation of a compact open subgroup of G, constructed from a β-extension and a cuspidal representation of a finite group. Secondly we show that two intertwining cuspidal types of G are up to equivalence conjugate under some ele- ment of G. [11E57][11E95][20G05][22E50] 1. Introduction This work is the third part in a series of three papers, the first two being [35] and [37]. Let F be a non-Archimedean local field with odd residue characteristic p. The construc- tion and classification of cuspidal irreducible representation complex or l-modular of the set of rational points G(F) of a reductive group G defined over F has already been success- fully studied for general linear groups (complex case: [11] Bushnell{Kutzko, [31], [4], [32] Broussous{Secherre{Stevens; modular case: [44] Vigneras, [23] Minguez{Sech´erre)and for p-adic classical groups ([42] Stevens, [21] Kurinczuk{Stevens, [20] Kurinczuk{Stevens joint with the author). In this paper we are generalizing from p-adic classical groups to their quaternionic forms. Let us mention [46] Yu, [15], [16] Fintzen and [19] Kim for results over reductive p-adic groups in general.
    [Show full text]
  • Supercuspidal Representations of Finite Reductive Groups
    JOURNAL OF ALGEBRA 184, 839]851Ž. 1996 ARTICLE NO. 0287 Supercuspidal Representations of Finite Reductive Groups Gerhard Hiss IWR der Uni¨ersitatÈ Heidelberg, Im Neuenheimer Feld 368, D-69120 Heidelberg, Germany View metadata, citation and similar papersCommunicated at core.ac.uk by Michel Broue brought to you by CORE provided by Elsevier - Publisher Connector Received January 9, 1995 DEDICATED TO MY PARENTS 1. INTRODUCTION A powerful tool in representation theory of finite groups of Lie type is Harish]Chandra induction. It is a method to construct, in a systematic way, representations for the group from those ofŽ. split Levi subgroups. Let G be a finite group of Lie type. An irreducible representationŽ of finite degree over a field containing the <<G th roots of unity. of G is called cuspidal if it does not occur as a subrepresentation of a Harish]Chandra induced representation of a proper Levi subgroup. In order to find all irreducible representations of G, one has to construct the cuspidal repre- sentations of all Levi subgroups first. Then one has to find all irreducible subrepresentations of Harish]Chandra induced cuspidal representations. Some of the cuspidal representations may occur as subquotients of Harish]Chandra induced representations. Following Vigneras, we call an irreducible representation of G super- cuspidal if it does not occur as a subquotient of a Harish]Chandra induced representation of a proper Levi subgroup. Thus, in particular, a supercuspidal representation is cuspidal. If the characteristic of the under- lying field does not divide the order of G, all representations of G are semisimple, so each subquotient is also a subrepresentation.
    [Show full text]
  • Arxiv:2102.03100V1 [Math.NT]
    L The special values of the standard -functions for GSp2n × GL1 Shuji Horinaga, Ameya Pitale, Abhishek Saha, Ralf Schmidt Abstract We prove the expected algebraicity property for the critical L-values of character twists of the standard L-function associated to vector-valued holomorphic Siegel cusp forms of archimedean type (k1, k2,...,kn), where kn ≥ 2n + 2 and all ki are of the same parity. For the proof, we use an explicit integral representation to reduce to arithmetic properties of differential operators on vector-valued nearly holomorphic Siegel cusp forms. We establish these properties via a representation-theoretic approach. Contents 1 Introduction 1 1.1 Notation........................................ 4 2 Lie algebra action and differential operators 5 2.1 Basics on the Lie algebra of Sp2n(R)......................... 6 2.2 Generators for the center of the universal enveloping algebra ........... 7 2.3 Differentialoperators. ..... 8 2.4 The relationship between elements of U(gC) and differential operators . 9 2.5 Scalar and vector-valued automorphic forms . ........... 10 3 Nearly holomorphic Siegel modular forms 14 3.1 Definitionsandbasicproperties . ........ 14 3.2 Nearly holomorphic modular forms and (p−)-finite automorphic forms . 15 3.3 Action of Aut(C).................................... 18 3.4 Arithmeticity of certain differential operators on nearly holomorphic cusp forms . 24 4 Special values of L-functions 28 arXiv:2102.03100v1 [math.NT] 5 Feb 2021 4.1 Automorphicrepresentations . ....... 28 4.2 Classicalandadeliccuspforms . ....... 29 4.3 Eisensteinseries ................................ .... 32 4.4 Algebraicity of critical L-values............................ 32 Bibliography 34 1 Introduction The arithmetic of special values of L-functions is of great interest in modern number theory.
    [Show full text]
  • Whittaker–Fourier Coefficients of Cusp Forms
    WHITTAKER{FOURIER COEFFICIENTS OF CUSP FORMS ZHENGYU MAO Abstract. We report on the work with E. Lapid about a conjecture relating Whittaker- Fourier coefficients of cusp forms to special values of L−functions. Contents 1. Introduction 1 2. Statement of the conjecture 2 3. Some known cases 5 4. Case of classical group and metaplectic group: automorphic descent 6 References 10 1. Introduction Let G be a quasi-split reductive group over a number field F and A the ring of adeles of F . Let B be a Borel subgroup of G defined over F , N the unipotent radical of B and fix a non-degenerate character N of N(A), trivial on N(F ). For a cusp form ' of G(F )nG(A) we consider the Whittaker{Fourier coefficient Z N −1 −1 W(') = W (') := (vol(N(F )nN(A))) '(n) N (n) dn: N(F )nN(A) If π is an irreducible cuspidal representation then W, if non-zero, gives a realization of π in the space of Whittaker functions on G(A), which by local multiplicity one depends only on π as an abstract representation. It therefore provides a useful tool for understanding π, both computationally and conceptually. It is natural to study the size of W('). In [LM15], inspired by the work of Ichino-Ikeda [II10], with E. Lapid we made a conjecture relating the size of W(') with a special L−value of π. In this note, we report on the work with Lapid on this conjecture. In modular form language, the notion of Whittaker-Fourier coefficient is roughly that of Fourier coefficient of a modular form.
    [Show full text]
  • Max-Planck-Institut Für Mathematik Bonn
    Max-Planck-Institut für Mathematik Bonn Non-vanishing square-integrable automorphic cohomology classes - the case GL(2) over a central division algebra by Joachim Schwermer Max-Planck-Institut für Mathematik Preprint Series 2020 (48) Date of submission: September 10, 2020 Non-vanishing square-integrable automorphic cohomology classes - the case GL(2) over a central division algebra by Joachim Schwermer Max-Planck-Institut für Mathematik Faculty of Mathematics Vivatsgasse 7 University Vienna 53111 Bonn Oskar-Morgenstern-Platz 1 Germany 1090 Vienna Austria MPIM 20-48 1 NON-VANISHING SQUARE-INTEGRABLE AUTOMORPHIC 2 COHOMOLOGY CLASSES - THE CASE GL(2) OVER A CENTRAL 3 DIVISION ALGEBRA 4 JOACHIM SCHWERMER Abstract. Let k be a totally real algebraic number field, and let D be a central divi- sion algebra of degree d over k. The connected reductive algebraic k-group GL(2;D)=k has k-rank one; it is an inner form of the split k-group GL(2d)=k. We construct auto- morphic representations π of GL(2d)=k which occur non-trivially in the discrete spec- trum of GL(2d; k) and which have specific local components at archimedean as well as non-archimedean places of k so that there exist automorphic representations π0 of 0 GL(2;D)(Ak) with Ξ(π ) = π under the Jacquet-Langlands correspondence. These re- quirements depend on the finite set VD of places of k at which D does not split, and on 0 the quest to construct representations π of GL(2;D)(Ak) which either represent cusp- idal cohomology classes or give rise to square-integrable classes which are not cuspidal, that is, are eventually represented by a residue of an Eisenstein series.
    [Show full text]
  • Arxiv:1404.0861V1 [Math.RT] 3 Apr 2014 St Td the Study to Is Rbe Foei Neetdi Opeecaatrinformat Character Complete in Interested Understood
    NOTES ON REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE DIPENDRA PRASAD These are the notes of some lectures given by the author for a workshop held at TIFR in December, 2011, giving an exposition of the Deligne-Lusztig theory. The aim of these lectures was to give an overview of the subject with several examples without burdening them with detailed proofs of the theorems involved for which we refer to the original paper of Deligne and Lusztig, as well as the beautiful book of Digne and Michel on the subject. The author thanks Shripad M. Garge for writing and texing the first draft of these notes. Contents 1. Preliminaries on Algebraic groups 2 2. Classical groups and tori 3 3. Some Exceptional groups 5 4. Basic notions in representation theory 6 5. The Deligne-Lusztig theory 7 6. Unipotent cuspidal representations 9 7. The Steinberg representation 9 8. Dimension of the Deligne-Lusztig representation 10 9. The Jordan decomposition of representations 11 10. Exercises 12 11. References 13 arXiv:1404.0861v1 [math.RT] 3 Apr 2014 The Deligne-Lusztig theory constructs certain (virtual) representations of G(Fq) denoted as RG(T, θ) where G is a connected reductive algebraic group defined over Fq, T is a maximal torus in G defined ∼ × over F and θ is a character of T (F ), θ : T (F ) C× Q . The representation R (T, θ) is called q q q → −→ l G the Deligne-Lusztig induction of the character θ of T (Fq) to G(Fq), generalizing parabolic induction with which it coincides when T is a maximally split maximal torus in G.
    [Show full text]
  • SUPERCUSPIDAL REPRESENTATIONS of Glnpfq DISTINGUISHED by a GALOIS INVOLUTION
    SUPERCUSPIDAL REPRESENTATIONS OF GLnpFq DISTINGUISHED BY A GALOIS INVOLUTION by Vincent Sécherre Abstract.— Let F{F0 be a quadratic extension of non-Archimedean locally compact fields of resi- dual characteristic p ‰ 2, and let σ denote its non-trivial automorphism. Let R be an algebraically closed field of characteristic different from p. To any cuspidal representation π of GLnpFq, with coef- ficients in R, such that πσ » π_ (such a representation is said to be σ-selfdual) we associate a quadra- tic extension D{D0, where D is a tamely ramified extension of F and D0 is a tamely ramified exten- ˆ sion of F0, together with a quadratic character of D0 . When π is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for π to be GLnpF0q-distinguished. When the charac- ˆ teristic ` of R is not 2, denoting by ω the non-trivial R-character of F0 trivial on F{F0-norms, we prove that any σ-selfdual supercuspidal R-representation is either distinguished or ω-distinguished, but not both. In the modular case, that is when ` ¡ 0, we give examples of σ-selfdual cuspidal non- supercuspidal representations which are not distinguished nor ω-distinguished. In the particular case where R is the field of complex numbers, in which case all cuspidal representations are supercuspi- dal, this gives a complete distinction criterion for arbitrary complex cuspidal representations, as well as a purely local proof, for cuspidal representations, of the dichotomy and disjunction theorem due to Kable and Anandavardhanan–Kable–Tandon, when p ‰ 2.
    [Show full text]
  • On the Computation of Local Components of a Newform 3
    ON THE COMPUTATION OF LOCAL COMPONENTS OF A NEWFORM DAVID LOEFFLER AND JARED WEINSTEIN 1. Introduction 1.1. The problem. Let f be a cuspidal newform for Γ1(N) with weight k 2 and character ε. There are well-established methods for computing such forms≥ using modular symbols, see [Ste07]. Let πf be the corresponding automorphic representa- tion of the ad`ele group GL2(AQ). (As there are numerous possible normalisations, we shall recall the construction in section 2.1 below.) This representation decom- poses as a restricted tensor product over all places v of Q: πf = πf,v, v O where πf,v is an irreducible admissible representation of GL2(Qv). The isomor- phism class of the infinite component πf,∞ is determined by the weight k alone: it is the unique discrete series subrepresentation of the representation constructed via unitary induction from the character k/2 t1 t1 k ∗ sgn(t1) t2 7→ t 2 of the Borel subgroup of GL2(R) [Gel75, prop. 5.21]. Similarly, the component πf,p for a prime p ∤ N is the unramified principal series with Satake parameters equal 2 k−1 to the roots of the polynomial X apX + ε(p)p . We are led naturally to the following problem: − Problem 1.1. Compute the local components π for each prime p N. f,p | Problem 1.1 may be recast from an arithmetic perspective as a question about the local behavior of Galois representations. Suppose the Fourier coefficients of f arXiv:1008.2796v1 [math.NT] 17 Aug 2010 lie in a number field E.
    [Show full text]
  • GEOMETRY of DUAL SPACES of REDUCTIVE GROUPS (Non-Archimedean Case)
    GEOMETRY OF DUAL SPACES OF REDUCTIVE GROUPS (non-archimedean case) MARKO TADIC Department of Mathematics, University of Zagreb, P.O. Box 187, 41001 Zagreb, Yugoslavia Introduction J. M. G. Fell introduced in [ 1O] the notion of the non-unitary dual space of a locally compact group. By his definition, it is a topological space. The main part of that paper deals with the basic properties of the topology of non-unitary dual space. This paper can be considered mainly as a continuation of such investi- gations, in the case of a p-adic reductive group G. We write down proofs of the basic properties of this topological space for such G. Since in the last twenty years after the appearance of [ 10] the topology of non-unitary dual has not attracted much attention, we begin with a longer introduction explaining motivations and reasons for writing this paper. Let G be a connected reductive group over a non-archimedean local field F. The set of all equivalence classes of topologically irreducible unitary represen- tations of G is denoted by G. The set d is in a natural way a topological space. The topology may be described in terms of approximation of matrix coefficients. In the study of some topological space a first reduction may be to decompose it into disjoint union of open and closed subsets. In [26] such a decomposition of G was obtained. One could see easily that the decomposition in [26] was not the finest possible of such type. This author realized that the right setting for the decompo- sition in [26] of (~ is not G but another topological space.
    [Show full text]