Local Theta Correspondence: the Basic Theory 3
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Rank One Reducibility for Metapletic Groups Via Theta Correspondence
Rank one reducibility for metapletic groups via theta correspondence Marcela Hanzer, Goran Mui´c Abstract We calculate reducibility for the representations of metaplectic groups induced from cuspidal representations of maximal parabolic subgroups via theta correspondence, in terms of the analogous repre- sentations of the odd orthogonal groups. We also describe the lifts of all relevant subquotients. 1 Introduction In this paper we study rank-one reducibility for the non{trivial double Sp^(n) cover of symplectic group Sp(n) over a non{Archimedean local field of charac- teristic different than two using theta correspondence. This paper combined with [8] is a fundamental step in a systematic study of smooth complex rep- resentations of metaplectic groups. We expect application in the theory of automorphic forms where metaplectic groups play a prominent role. We recall that the group Sp^(n) is not a linear algebraic group. Thus, it is not in the framework of the usual theory for p-adic groups. Neverthe- less, some basic algebraic facts [8] are true here. More precisely, based on a fundamental work of Bernstein and Zelevinsky ([2], [3], [26]) we checked that the basic notions of the representation theory of p{adic groups hold for metaplectic groups (some of that is already well{known from the previous works of Kudla [10], [9] and the book [15]). As usual, a parabolic subgroup of Sp^(n) is the preimage Pe of a parabolic subgroup P of Sp(n). If we write P = MN for a Levi decomposition, then the unipotent radical N lifts to Sp^(n): So, we have a decomposition Pe = MNf . -
View This Volume's Front and Back Matter
http://dx.doi.org/10.1090/gsm/039 Selected Titles in This Series 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2001 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2001 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. -
Symmetry Breaking Operators for Dual Pairs with One Member Compact M Mckee, Angela Pasquale, T Przebinda
Symmetry breaking operators for dual pairs with one member compact M Mckee, Angela Pasquale, T Przebinda To cite this version: M Mckee, Angela Pasquale, T Przebinda. Symmetry breaking operators for dual pairs with one member compact. 2021. hal-03293407 HAL Id: hal-03293407 https://hal.archives-ouvertes.fr/hal-03293407 Preprint submitted on 21 Jul 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. SYMMETRY BREAKING OPERATORS FOR DUAL PAIRS WITH ONE MEMBER COMPACT M. MCKEE, A. PASQUALE, AND T. PRZEBINDA Abstract. We consider a dual pair (G; G0), in the sense of Howe, with G compact acting on L2(Rn), for an appropriate n, via the Weil representation !. Let Ge be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation Π of G,e let Π0 be the corresponding irreducible unitary representation of Ge 0 in the Howe duality. 2 n The orthogonal projection onto L (R )Π, the Π-isotypic component, is the essentially 1 1 1 unique symmetry breaking operator in Hom (H ; H ⊗H 0 ). We study this operator GeGf0 ! Π Π by computing its Weyl symbol. -
R–Groups for Metaplectic Groups∗
R{groups for metaplectic groups∗ Marcela Hanzery1 1Department of Mathematics, University of Zagreb, Croatia Abstract In this short note, we completely describe a parabolically induced representation Sp^(2n;F ) IndP (σ); in particular, its length and multiplicities. Here, Sp^(2n; F ) is a p{adic metaplectic group, and σ is a discrete series representation of a Levi subgroup of P: A multiplicity one result follows. 1 Introduction A knowledge of R{groups (in all their variants{classical, Arthur, etc.) is very important in understanding the representation theory of reductive p{adic groups, especially their unitary duals. Let G be a reductive p{adic group with a parabolic subgroup P = MN where M is a Levi subgroup. Assume σ is a discrete series (complex) representation of M: Then, it is G important to understand how the representation IndP (σ) reduces: whether it is reducible, and if it is reducible, whether all the irreducible subquotients appear with multiplicity one. These questions are answered by knowing the structure of the commuting algebra C(σ); G i.e., the intertwining algebra of IndP (σ): It is already known from the work of Casselman that the dimension of C(σ) is bounded by the cardinality of W (σ); the subgroup of the Weyl group of G fixing the representation σ: The precise structure of C(σ) is given by a certain subgroup of W (σ); called the R{group of σ: This approach to understanding the G structure of IndP (σ) goes back to the work of Knapp and Stein on real groups and principal series representations. -
On First Occurrence in the Local Theta Correspondence
On First Occurrence in the Local Theta Correspondence Stephen S. Kudla∗ and Stephen Rallis† Abstract. This paper discusses a conservation conjecture for the first oc- currence indices. Such indices record the first occurrence of an irreducible admissible representation π of a fixed group G in the local theta correspon- dence as the second member of a dual pair (G, G) runs over the isometry groups in a Witt tower. For example, for a fixed nonarchimedean local field F and for a fixed quadratic character χ of F ×, there are two Witt towers + − ± {Vm } and {Vm }, where the quadratic space Vm has dimension m,character ± ± χ and Hasse invariant ±1. If mχ (π)isthe dimension of the space Vm for which the irreducible admissible representation π of Spn(F ) first occurs in the ± theta correspondence for the dual pair (Spn(F ),O(Vm )), then we conjecture that + − mχ (π)+mχ (π)=4n +4. We prove this conjecture in many cases, in particular, when π is a supercus- pidal representation. 2000 Mathematics Subject Classification: primary 11F27; secondary 22E50. 0. Introduction A basic step in understanding the local theta correspondence is the descrip- tion of the behavior of the correspondence as one group in the reductive dual pair varies in a Witt tower, [18]. The combination of this Witt tower tech- nique, with the method of ‘doubling’, [4], [18], and the local ‘doubling’ zeta integrals, [16], [17], lies behind much of our qualitative understanding of the correspondence. In this paper1,inthe case of a nonarchimedean local field, ∗Partially supported by NSF Grant DMS-0200292 and by a Max-Planck Research Prize from the Max-Planck Society and Alexander von Humboldt Stiftung. -
Dual Pairs in the Pin-Group and Duality for the Corresponding Spinorial
DUAL PAIRS IN THE PIN-GROUP AND DUALITY FOR THE CORRESPONDING SPINORIAL REPRESENTATION CLÉMENT GUÉRIN, GANG LIU, AND ALLAN MERINO Abstract. In this paper, we give a complete picture of Howe correspondence for the setting (O(E, b), Pin(E, b), Π), where O(E, b) is an orthogonal group (real or complex), Pin(E, b) is the two-fold Pin-covering of O(E, b), and Π is the spinorial representation of Pin(E, b). More pre- cisely, for a dual pair (G, G′) in O(E, b), we determine explicitly the nature of its preimages (G, G′) in Pin(E, b), and prove that apart from some exceptions, (G, G′) is always a dual pair in e f e f Pin(E, b); then we establish the Howe correspondence for Π with respect to (G, G′). e f Contents 1. Introduction 1 2. Preliminaries 3 3. Dual pairs in the Pin-group 6 3.1. Pull-back of dual pairs 6 3.2. Identifyingtheisomorphismclassofpull-backs 8 4. Duality for the spinorial representation 14 References 21 1. Introduction The first duality phenomenon has been discovered by H. Weyl who pointed out a correspon- dence between some irreducible finite dimensional representations of the general linear group GL(V) and the symmetric group Sk where V is a finite dimensional vector space over C. In- d deed, considering the joint action of GL(V) and Sd on the space V⊗ , we get the following decomposition: arXiv:1907.09093v1 [math.RT] 22 Jul 2019 d λ V⊗ = M λ σ , [ ⊗ (Vλ,λ) GL(V) ∈ π [ where GL(V)π is the set of equivalence classes of irreducible finite dimensional representations d of GL(V) such that Hom (V , V⊗ ) , 0 and σ is an irreducible representation of S . -
Singular Theta Lifts and Near-Central Special Values of Rankin-Selberg L-Functions
Singular theta lifts and near-central special values of Rankin-Selberg L-functions Luis E. Garcia Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2013 c 2013 Luis E. Garcia All Rights Reserved ABSTRACT Singular theta lifts and near-central special values of Rankin-Selberg L-functions Luis E. Garcia In this thesis we study integrals of a product of two automorphic forms of weight 2 on a Shimura curve over Q against a function on the curve with logarithmic singularities at CM points obtained as a Borcherds lift. We prove a formula relating periods of this type to a near-central special value of a Rankin-Selberg L-function. The results provide evidence for Beilinson's conjectures on special values of L-functions. Table of Contents 1 Introduction 1 1.1 Statement of the main theorem . 1 1.2 Outline of the thesis . 3 1.3 Beilinson's conjectures . 5 1.4 The case of Rankin-Selberg L-functions . 10 1.5 An example . 12 1.6 Notations and measures . 14 2 Theta Correspondence for (Sp4;O4) 18 2.1 Four-dimensional quadratic spaces and their orthogonal groups . 18 2.2 Yoshida lifts . 20 2.3 Weil representation . 21 2.4 Non-archimedean correspondence . 21 2.5 Archimedean correspondence . 24 2.6 Special Schwartz forms . 27 2.7 Siegel-Weil formula . 30 3 Singular theta lifts 32 3.1 Borcherds lifts . 32 3.2 Harmonic Whittaker forms . 38 4 Local Zeta integrals 42 4.1 Doubling Integrals . -
Theta Liftings on Higher Covers of Symplectic Groups
Theta liftings on higher covers of symplectic groups Author: Spencer Leslie Persistent link: http://hdl.handle.net/2345/bc-ir:107937 This work is posted on eScholarship@BC, Boston College University Libraries. Boston College Electronic Thesis or Dissertation, 2018 Copyright is held by the author, with all rights reserved, unless otherwise noted. Theta liftings on higher covers of symplectic groups Spencer Leslie A dissertation submitted to the Faculty of the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy Boston College Morrissey College of Arts and Sciences Graduate School May 2018 c Copyright 2018 Spencer Leslie Theta liftings on higher covers of symplectic groups Spencer Leslie Thesis advisor: Solomon Friedberg We study a new lifting of automorphic representations using the theta representa- tion Θ on the 4-fold cover of the symplectic group, Sp2r(A). This lifting produces the first examples of CAP representations on higher degree metaplectic covering groups. Central to our analysis is the identification of the maximal nilpotent orbit associated to Θ. We conjecture a natural extension of Arthur's parameterization of the discrete spectrum to Sp2r(A). Assuming this, we compute the effect of our lift on Arthur parameters and show that the parameter of a representation in the image of the lift is non-tempered. We conclude by relating the lifting to the dimension equation of Ginzburg to predict the first non-trivial lift of a generic cuspidal representation of Sp2r(A). Contents 1 Introduction1 1.1 Theta functions and liftings.......................1 1.2 Main Results...............................6 2 Notations and Preliminaries 13 2.1 Subgroups and characters....................... -
Sylow P -Subgroups of the Classical Groups Over Finite Fields with Characteristic Prime to P
SYLOW /--SUBGROUPS OF THE CLASSICAL GROUPS OVER FINITE FIELDS WITH CHARACTERISTIC PRIME TO p A. J. WEIR If Sn is a Sylow /--subgroup of the symmetric group of degree pn, then any group of order pn may be imbedded in 5„. We may express Sn as the complete product1 C o C o • • • o C of « cyclic groups of order p and the purpose of this paper is to show that any Sylow p- subgroup of a classical group (see §1) over the finite field GF(q) with q elements, where (q, p) = 1, is expressible as a direct product of basic subgroups 5„= C o C o • • • o C (n factors), where C is cyclic of order pT. (We assume always that p7£2.) Since C may be imbedded in ST, we see that Sn is imbedded in Sn+r-i in a particularly simple way. The above r is defined by the equation q*— \ =pT * where g* is the first power of q which is congruent to 1 mod p and * denotes some unspecified number prime to p. The case r = 1 is therefore of frequent occurrence, and then clearly Sn=Sn. Professor Philip Hall was my research supervisor in Cambridge (England) during the years 1949-1952 and it is a pleasure to acknowl- edge here my indebtedness to him for his generous encouragement. 1. We shall refer to the following groups as the Classical Groups:* I. The general linear group GL(n, q) is the group of all nonsingular (nXn) matrices with coefficients in GF(q). The order of GL(n, q) is qn(n-l)l2fq _ 1)(?2 _ J) . -
Two Topics on Local Theta Correspondence Ma Jia
TWO TOPICS ON LOCAL THETA CORRESPONDENCE MA JIA JUN (B.Sc., Soochow University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2012 Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Ma Jia Jun 20 February 2013 ACKNOWLEDGEMENTS I would like to take this opportunity to acknowledge and thank those who made this work possible. I would like to express my deep gratitude to Prof. Chengbo Zhu, my supervisor for his supervision and constant support. Prof. Zhu leads me to this exciting research area, proposes interesting questions and always provides illuminating suggestions to me during my study. I am sincerely grateful to Prof. Hung Yean Loke, who have spent enor- mous of time in patient discussion with me and given me lots of inspiring advices. In the collaboration with Prof. Loke, I learnt many mathematics from him. I am profoundly indebted to Prof. Soo Teck Lee, who launched instructive seminars which deeply influenced this work. I express my sincere thanks to Prof. CheeWhye Chin and Prof. De-Qi Zhang, who patiently explained lots of concepts in algebraic geometry to me. I also would like to thank Prof. Michel Brion, Prof. Wee Teck Gan, Prof. Roger Howe, Prof. Jingsong Huang, Prof. Kyo Nishiyama, Prof. Gordan Savin and Prof. Binyong Sun, for their stimulating conversations and suggestions. -
The Theta Correspondence Over R
The Theta Correspondence over R Jeffrey Adams May 3, 2007 1 Introduction These notes were written for the workshop on the Theta correspondence held at the University of Maryland, May 1994. They have circulated informally for many years. Except for minor changes and some improvements to the references they have not been changed. There have been too many develop- ments in the field since 1994 to try to bring them up to date. This is dedicated to Roger Howe, who introduced me to representation theory in graduate school in 1977. 2 Fock model: complex Lie algebra The Fock model is the algebraic oscillator representation: it is the (g; K) module of the oscillator representation of Sp(2n; R). The basic references e are [6],[7], and [47]. See also [11],[17] and [18]. The presentation here is due f to Steve Kudla (unpublished); a similar treatment is found in [49], as well as [51]. In this section we define the oscillator representation of the complex Heisenberg and symplectic Lie algebras. Let be a non{trivial linear form on C, i.e. fix λ 2 C× and let (z) = λz (z 2 C). Definition 2.1 Let W be a vector space with a symplectic form h; i, possibly degenerate. Let Ω(W ) = T (W )=I 1 where T is the tensor algebra of W and I is the two{sided ideal generated by elements of the form v ⊗ w − w ⊗ v − hv; wi (v; w 2 W ): This is an associative algebra, sometimes referred to as the quantum algebra. For example if hv; wi = 0 for all v; w, then Ω(W ) is isomorphic to the polynomial algebra on W . -
Representations and Invariants of the Classical Groups, by Roe Goodman and Nolan R
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 4, Pages 533{538 S 0273-0979(99)00795-8 Article electronically published on July 28, 1999 Representations and invariants of the classical groups, by Roe Goodman and Nolan R. Wallach, Cambridge Univ. Press, Cambridge, 1998, xvi + 685 pp., $100.00, ISBN 0-521-58273-3, paperback, $39.95, ISBN 0-521-66348-2 This 685-page text by Goodman and Wallach constitutes a carefully organized exposition, not indeed of the entire finite-dimensional representation-theory over R or C of the classical groups (which would be hard to imagine in a single text three times the size of this one), but rather of a selection of some key topics therein. This selection of topics, especially as concerns invariant-theory, seems (as Goodman and Wallach indicate in the preface to their text) to have been in many ways inspired by the contents of H. Weyl’s The Classical Groups, Their Invariants and Representations [31]. Approximately half of the text under review is purely introductory; the remaining half then utilizes this introductory material to develop the main themes. The introductory material, covered in a total of approximately 280 pages in the first three chapters and in four appendices, introduces the reader to some necessary topics in elementary algebraic and differential geometry and to such basic concepts of representation theory as: linear algebraic groups and Lie groups, Lie algebras associated to such, Jordan decomposition, maximal tori, roots and weights, real forms of classical groups, the PBW theorem, invariant integration over compact Lie groups, etc.