DOCSLIB.ORG

Math 679: Automorphic Forms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 679: Automorphic Forms MATH 679: AUTOMORPHIC FORMS LECTURES BY PROF. TASHO KALETHA; NOTES BY ALEKSANDER HORAWA These are notes from Math 679 taught by Professor Tasho Kaletha in Winter 2019, LATEX'ed by Aleksander Horawa (who is the only person responsible for any mistakes that may be found in them). This version is from September 26, 2019. Check for the latest version of these notes at http://www-personal.umich.edu/~ahorawa/index.html If you find any typos or mistakes, please let me know at [email protected]. Contents 1. Introduction: Review of modular forms1 2. Overview of harmonic analysis on LCA groups 13 3. Harmonic analysis on local fields and adeles 19 4. Tate's thesis 25 5. Artin L-functions 42 6. Non-abelian class field theory? 59 7. Automorphic representations of SL2(R) 61 8. Automorphic representations of SL2(Qp) 81 9. Automorphic representations of GL(2; A) 106 1. Introduction: Review of modular forms 1.1. Automorphy and functional equations of L-functions. Let H = fz 2 C j Im(z) > 0g be the upper half plane. It has an action of the Lie group SL2(R) of 2 × 2 matrices with R-coefficients and determinant 1 by a b az + b z = : c d cz + d Definition 1.1. A holomorphic modular form of weifght k 2 Z is a holomorphic function f : H ! C Date: September 26, 2019. 1 2 TASHO KALETHA satisfying a b • f(γz) = (cz + d)kf(z) for all γ = 2 SL ( ), c d 2 Z • f is bounded on domains of the form fz 2 H j Im(z) > C > 0g. The second condition is equivalent to f being holomorphic at i1. Remark 1.2. The group SL2(Z) is generated by 1 1 1 0 and ; 0 1 1 1 and by 1 1 0 1 and : 0 1 −1 0 1 1 In particular, taking γ = , we get 0 1 f(z + 1) = f(z); so f has a Fourier expansion X n 2πiz anq for q = e : n2Z Then the second condition is also equivalent to saying an = 0 for n < 0: By Remark 1.2, the first condition is equivalent to 1 f(z + 1) = f(z) and f − = (−z)kf(z): z How can we generalize this? (1) We can replace SL2(Z) by a subgroup (usually of finite index). Definition 1.3. For N 2 N, let Γ(N) = ker(SL2(Z) ! SL2(Z=NZ)): A subgroup Γ ⊆ SL2(Z) is congruence if it contains Γ(N) for some N. Example 1.4. The subgroup Γ(2) is generated by 1 2 1 0 −1 0 ; ; : 0 1 2 1 0 −1 Let ∆ ⊆ Γ(2) be the subgroup generated by these. Notice that ∆ is normal in SL2(Z). Consider Γ(2)=∆. An element of Γ(2)=∆ can be represented as an alternating 1 1 1 0 word in , . Compute to check that the smallest length of a word 0 1 −1 1 contained in Γ(2) has length 6 and it is equal to −1 0 2 ∆: 0 −1 MATH 679: AUTOMORPHIC FORMS 3 1 2 0 1 Example 1.5. The subgroup Γ generated by , called the theta group. 0 1 −1 0 (2) We can allow half-integralp weight. On thep slices complex place C n R<0. We have an analytic map z 7! z determined by Re( z) p> 0. p This can be extended non-continuously by −r = i r for r 2 R>0. k We can try to define, for k 2 Z, a modular form of weight 2 by requiring that f(γz) = (cz + d)k=2f(z): We will write ν(γ; z) = (cz + d). Remark 1.6. We have that ν(γδ; z) = ν(γ; δz)ν(δ; z). This is no longer true for 1 ν(γ; z) 2 . It is only true up to a ± sign. Therefore, we introduce a multiplicator system: µ:Γ ! µ1(C) such that k µ(γ) · ν(γ; z) 2 has the desired behavior. Example 1.7. The theta multiplicator system is defined by 1 2 0 1 µ = 1; µ = eπi=4: 0 1 −1 0 Example 1.8. The Jacobi theta function X 2 θ(z) = eπin z n2Z 1 is a modular forms of weight 2 for Γ the theta group. This converges locally uniformly. Take a domain fIm(z) > C > 0g. If z = x + iy is in this domain, then 2 2 2 2 πin z πin x −πn y −πn C e = e · e < e is bounded uniformly. 1 2 0 1 To check automorphy, note that γ = is obvious. The case γ = is 0 1 −1 0 interesting. Z 2πixy Fourier analysis. Let f : R ! C and f^ = f(x)e dx. We have the inversion R formula ^ f^ = f(−x) and the Poisson summation formula X X f(n) = f^(n): n2Z n2Z 4 TASHO KALETHA The Gaussian f(x) = e−πx2 is its own Fourier transform: Z f^(y) = e−πx2+2πixydx R Z = e−π(x2−2ixy+(iy)2)e−πy2 dx R Z = e−πy2 e−π(x−iy)2 dx: R We claim that the last integral is equal to 1. (a) The contour integral of e−πz2 over the region R which is a rectangle with corner −T; T; iy − T; iy + T is 0 because the function is holomorphic. Therefore, letting T ! 1, Z Z e−π(x−iy)2 dx = e−πx2 dx: R R (b) Rescale to see that Z 2 1 Z 2 e−πx dx = p e−x dx: R π R (c) Take square to see that Z 2 Z e−x2 dx = e−x2−y2 dxdy Z 2π Z 1 = e−r2 rdrd' changing to polar coordinates 0 0 " 2 #1 e−r = 2π −2 0 = π: 1=2 Let ft(x) = f(xt ) for t 2 R>0. Note that 1 1 f^ (y) = f^(y=t) = f(y=t) t t1=2 t1=2 We want to show that 1 z 1=2 θ − = eπi=y · (−z)1=2θ(z) = θ(z): z i MATH 679: AUTOMORPHIC FORMS 5 z Since i is in the right half plane where the square root is analytic, both sides are analytic functions, so it is enough to take z = it, t 2 R>0. Then X θ(it) = ft(n) n2Z X ^ = ft(n) Poisson summation n2Z 1 X = f (n=t) t1=2 t n2Z 1 i = θ : t1=2 t Recall that the Riemann zeta function is defined as 1 X 1 ζ(s) = for Re(s) > 1: ns n=1 We want to show (1) ζ has an analytic continuation to C with a pole at s = 1, (2) ζ obeys a functional equation. Note that ζ is incomplete. We have the Euler product: Y ζ(s) = (1 − p−s)−1: p In hindsight, we are missing an Euler factor for p = 1. Define Λ(s) = π−s=2Γ(s=2)ζ(s); where Z 1 dt Γ(s) = tse−t 0 t is the Γ-function, defined for Re(s) > 0. Remark 1.9. This is a \Fourier transform" on (R×; ·) in place of (R; +). Indeed, normally we integrate the function against the character x 7! e2πixy of (R; +) with respect to the addition-invariant measure dt. Here, we integrate against the character t 7! ts of (R×; ·) dt with respect to the multiplication-invariant measure t . This is called a Mellin transform. We have X 2 θ(z) = eπin z n2Z 1 X 2 = 1 + 2 eπin z : n=1 | {z } w(z) 6 TASHO KALETHA Let us apply the Mellin transform to t 7! w(it): Z 1 Z 1 1 dt X 2 dt w(it)ts = e−πn tts t t 0 0 n=1 1 Z 1 X 2 dt = e−πn tts t n=1 0 1 s X Z 1 t dt = e−t πn2 t n=1 0 1 X 1 Z 1 dt = e−tts (πn2)s t n=1 0 = π−sζ(2s)Γ(s) = Λ(2s): From the automorphy of θ, we get 1 1 1 w − = θ − − 1 z 2 z 1 z 1=2 = θ(z) − 1 2 i 1 1 = (−iz)1=2w(z) + (−iz)1=2 − : 2 2 Therefore, i 1 1 w = t1=2w(it) + t1=2 − : t z 2 Using this, we get that Z 1 dt Λ(2s) = w(it)ts 0 t Z 1 dt Z 1 dt = w(it)ts + w(it)ts 1 t 0 t Z 1 dt Z 1 i dt = w(it)ts + w t−s 1 t 1 t t Z 1 Z 1 s −s+ 1 dt 1 −s+1=2 −s dt = w(it)(t + t 2 ) + (t − t ) : 1 t 2 1 t We consider these two summands separately. The second summand is 1 Z 1 1 t1=2−s 1 1 t−s 1 1 1 1 t−1=2−s − t−1−sdt = − = + = : 2 2 1 2 −s 1 − 2s 2s 2s(1 − 2s) 1 2 − s 1 1 Let us now consider the first summand. We claim that jw(it)j decays exponentially as t ! 1. Recall that writing 1 X n θ(z) = anq ; n=0 MATH 679: AUTOMORPHIC FORMS 7 we get that 1 1 1 X 1 X w(z) = a qn = q a qn : 2 n 2 n+1 n=1 n=0 | {z } bounded around 0 Therefore, the first integral converges for any s, and gives an analytic function f(s), invariant 1 under s 7! 2 − s Altogether, we conclude that 1 1 (1) Λ(2s) = 1−2s + 2s + f(s) |{z} analytic (2) Λ(1 − s) = Λ(s).
Load more

Recommended publications
  • Automorphic Forms on GL(2)
    Automorphic Forms on GL(2) Herve´ Jacquet and Robert P. Langlands Formerly appeared as volume #114 in the Springer Lecture Notes in Mathematics, 1970, pp. 1-548 Chapter 1 i Table of Contents Introduction ...................................ii Chapter I: Local Theory ..............................1 § 1. Weil representations . 1 § 2. Representations of GL(2,F ) in the non•archimedean case . 12 § 3. The principal series for non•archimedean fields . 46 § 4. Examples of absolutely cuspidal representations . 62 § 5. Representations of GL(2, R) ........................ 77 § 6. Representation of GL(2, C) . 111 § 7. Characters . 121 § 8. Odds and ends . 139 Chapter II: Global Theory ............................152 § 9. The global Hecke algebra . 152 §10. Automorphic forms . 163 §11. Hecke theory . 176 §12. Some extraordinary representations . 203 Chapter III: Quaternion Algebras . 216 §13. Zeta•functions for M(2,F ) . 216 §14. Automorphic forms and quaternion algebras . 239 §15. Some orthogonality relations . 247 §16. An application of the Selberg trace formula . 260 Chapter 1 ii Introduction Two of the best known of Hecke’s achievements are his theory of L•functions with grossen•¨ charakter, which are Dirichlet series which can be represented by Euler products, and his theory of the Euler products, associated to automorphic forms on GL(2). Since a grossencharakter¨ is an automorphic form on GL(1) one is tempted to ask if the Euler products associated to automorphic forms on GL(2) play a role in the theory of numbers similar to that played by the L•functions with grossencharakter.¨ In particular do they bear the same relation to the Artin L•functions associated to two•dimensional representations of a Galois group as the Hecke L•functions bear to the Artin L•functions associated to one•dimensional representations? Although we cannot answer the question definitively one of the principal purposes of these notes is to provide some evidence that the answer is affirmative.
    [Show full text]
  • REPRESENTATION THEORY WEEK 7 1. Characters of GL Kand Sn A
    REPRESENTATION THEORY WEEK 7 1. Characters of GLk and Sn A character of an irreducible representation of GLk is a polynomial function con- stant on every conjugacy class. Since the set of diagonalizable matrices is dense in GLk, a character is defined by its values on the subgroup of diagonal matrices in GLk. Thus, one can consider a character as a polynomial function of x1,...,xk. Moreover, a character is a symmetric polynomial of x1,...,xk as the matrices diag (x1,...,xk) and diag xs(1),...,xs(k) are conjugate for any s ∈ Sk. For example, the character of the standard representation in E is equal to x1 + ⊗n n ··· + xk and the character of E is equal to (x1 + ··· + xk) . Let λ = (λ1,...,λk) be such that λ1 ≥ λ2 ≥ ···≥ λk ≥ 0. Let Dλ denote the λj determinant of the k × k-matrix whose i, j entry equals xi . It is clear that Dλ is a skew-symmetric polynomial of x1,...,xk. If ρ = (k − 1,..., 1, 0) then Dρ = i≤j (xi − xj) is the well known Vandermonde determinant. Let Q Dλ+ρ Sλ = . Dρ It is easy to see that Sλ is a symmetric polynomial of x1,...,xk. It is called a Schur λ1 λk polynomial. The leading monomial of Sλ is the x ...xk (if one orders monomials lexicographically) and therefore it is not hard to show that Sλ form a basis in the ring of symmetric polynomials of x1,...,xk. Theorem 1.1. The character of Wλ equals to Sλ. I do not include a proof of this Theorem since it uses beautiful but hard combina- toric.
    [Show full text]
  • Matrix Coefficients and Linearization Formulas for SL(2)
    Matrix Coefficients and Linearization Formulas for SL(2) Robert W. Donley, Jr. (Queensborough Community College) November 17, 2017 Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 1 / 43 Goals of Talk 1 Review of last talk 2 Special Functions 3 Matrix Coefficients 4 Physics Background 5 Matrix calculator for cm;n;k (i; j) (Vanishing of cm;n;k (i; j) at certain parameters) Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 2 / 43 References 1 Andrews, Askey, and Roy, Special Functions (big red book) 2 Vilenkin, Special Functions and the Theory of Group Representations (big purple book) 3 Beiser, Concepts of Modern Physics, 4th edition 4 Donley and Kim, "A rational theory of Clebsch-Gordan coefficients,” preprint. Available on arXiv Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 3 / 43 Review of Last Talk X = SL(2; C)=T n ≥ 0 : V (2n) highest weight space for highest weight 2n, dim(V (2n)) = 2n + 1 ∼ X C[SL(2; C)=T ] = V (2n) n2N T X C[SL(2; C)=T ] = C f2n n2N f2n is called a zonal spherical function of type 2n: That is, T · f2n = f2n: Robert W. Donley, Jr. (Queensborough CommunityMatrix College) Coefficients and Linearization Formulas for SL(2)November 17, 2017 4 / 43 Linearization Formula 1) Weight 0 : t · (f2m f2n) = (t · f2m)(t · f2n) = f2m f2n min(m;n) ∼ P 2) f2m f2n 2 V (2m) ⊗ V (2n) = V (2m + 2n − 2k) k=0 (Clebsch-Gordan decomposition) That is, f2m f2n is also spherical and a finite sum of zonal spherical functions.
    [Show full text]
  • An Introduction to Lie Groups and Lie Algebras
    This page intentionally left blank CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 113 EDITORIAL BOARD b. bollobás, w. fulton, a. katok, f. kirwan, p. sarnak, b. simon, b. totaro An Introduction to Lie Groups and Lie Algebras With roots in the nineteenth century, Lie theory has since found many and varied applications in mathematics and mathematical physics, to the point where it is now regarded as a classical branch of mathematics in its own right. This graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups, to the theory of root systems, and classification of finite-dimensional representations of semisimple Lie algebras. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York at Stony Brook, the book includes numerous exercises and worked examples and is ideal for graduate courses on Lie groups and Lie algebras. CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit: http://www.cambridge.org/series/ sSeries.asp?code=CSAM Already published 60 M. P. Brodmann & R. Y. Sharp Local cohomology 61 J. D. Dixon et al. Analytic pro-p groups 62 R. Stanley Enumerative combinatorics II 63 R. M. Dudley Uniform central limit theorems 64 J.
    [Show full text]
  • Math.GR] 21 Jul 2017 01-00357
    TWISTED BURNSIDE-FROBENIUS THEORY FOR ENDOMORPHISMS OF POLYCYCLIC GROUPS ALEXANDER FEL’SHTYN AND EVGENIJ TROITSKY Abstract. Let R(ϕ) be the number of ϕ-conjugacy (or Reidemeister) classes of an endo- morphism ϕ of a group G. We prove for several classes of groups (including polycyclic) that the number R(ϕ) is equal to the number of fixed points of the induced map of an appropriate subspace of the unitary dual space G, when R(ϕ) < ∞. Applying the result to iterations of ϕ we obtain Gauss congruences for Reidemeister numbers. b In contrast with the case of automorphisms, studied previously, we have a plenty of examples having the above finiteness condition, even among groups with R∞ property. Introduction The Reidemeister number or ϕ-conjugacy number of an endomorphism ϕ of a group G is the number of its Reidemeister or ϕ-conjugacy classes, defined by the equivalence g ∼ xgϕ(x−1). The interest in twisted conjugacy relations has its origins, in particular, in the Nielsen- Reidemeister fixed point theory (see, e.g. [29, 30, 6]), in Arthur-Selberg theory (see, e.g. [42, 1]), Algebraic Geometry (see, e.g. [27]), and Galois cohomology (see, e.g. [41]). In representation theory twisted conjugacy probably occurs first in [23] (see, e.g. [44, 37]). An important problem in the field is to identify the Reidemeister numbers with numbers of fixed points on an appropriate space in a way respecting iterations. This opens possibility of obtaining congruences for Reidemeister numbers and other important information. For the role of the above “appropriate space” typically some versions of unitary dual can be taken.
    [Show full text]
  • Representation Theory
    M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23.
    [Show full text]
  • Schur Orthogonality Relations and Invariant Sesquilinear Forms
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 4, Pages 1211{1219 S 0002-9939(01)06227-X Article electronically published on August 29, 2001 SCHUR ORTHOGONALITY RELATIONS AND INVARIANT SESQUILINEAR FORMS ROBERT W. DONLEY, JR. (Communicated by Rebecca Herb) Abstract. Important connections between the representation theory of a compact group G and L2(G) are summarized by the Schur orthogonality relations. The first part of this work is to generalize these relations to all finite-dimensional representations of a connected semisimple Lie group G: The second part establishes a general framework in the case of unitary representa- tions (π, V ) of a separable locally compact group. The key step is to identify the matrix coefficient space with a dense subset of the Hilbert-Schmidt endo- morphisms on V . 0. Introduction For a connected compact Lie group G, the Peter-Weyl Theorem [PW] provides an explicit decomposition of L2(G) in terms of the representation theory of G.Each irreducible representation occurs in L2(G) with multiplicity equal to its dimension. The set of all irreducible representations are parametrized in the Theorem of the Highest Weight. To convert representations into functions, one forms the set of matrix coefficients. To undo this correspondence, the Schur orthogonality relations express the L2{inner product in terms of the unitary structure of the irreducible representations. 0 Theorem 0.1 (Schur). Fix irreducible unitary representations (π, V π), (π0;Vπ ) of G.Then Z ( 1 h 0ih 0i ∼ 0 0 0 0 u; u v; v if π = π ; hπ(g)u; vihπ (g)u ;v i dg = dπ 0 otherwise, G π 0 0 π0 where u; v are in V ;u;v are in V ;dgis normalized Haar measure, and dπ is the dimension of V π.
    [Show full text]
  • On a Class of Inverse Quadratic Eigenvalue Problem
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 235 (2011) 2662–2669 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam On a class of inverse quadratic eigenvalue problem Yongxin Yuan a,∗, Hua Dai b a School of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang 212003, PR China b Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China article info a b s t r a c t Article history: In this paper, we first give the representation of the general solution of the following inverse Received 25 September 2009 monic quadratic eigenvalue problem (IMQEP): given matrices Λ D diagfλ1; : : : ; λpg 2 Received in revised form 3 December 2009 p×p n×p C , λi 6D λj for i 6D j, i; j D 1;:::; p, X DTx1;:::; xpU 2 C , rank.X/ D p, and N both Λ and X are closed under complex conjugation in the sense that λ2j D λ2j−1 2 C, MSC: n n x2j D xN2j−1 2 C for j D 1;:::; l, and λk 2 R, xk 2 R for k D 2l C 1;:::; p, 65F18 2 C C D 15A24 find real-valued symmetric matrices D and K such that XΛ DXΛ KX 0: Then Q Q n×n O O we consider a best approximation problem: given D; K 2 R , find .D; K/ 2 SDK such Keywords: O O Q Q Q Q that k.D; K/ − .D; K/kW D min.D;K/2 k.D; K/ − .D; K/kW ; where k · kW is a weighted Quadratic eigenvalue problem SDK Frobenius norm and SDK is the solution set of IMQEP.
    [Show full text]
  • Random Lie Group Actions on Compact Manifolds
    The Annals of Probability 2010, Vol. 38, No. 6, 2224–2257 DOI: 10.1214/10-AOP544 c Institute of Mathematical Statistics, 2010 RANDOM LIE GROUP ACTIONS ON COMPACT MANIFOLDS: A PERTURBATIVE ANALYSIS1 By Christian Sadel and Hermann Schulz-Baldes Universit¨at Erlangen–N¨urnberg A random Lie group action on a compact manifold generates a discrete time Markov process. The main object of this paper is the evaluation of associated Birkhoff sums in a regime of weak, but suf- ficiently effective coupling of the randomness. This effectiveness is expressed in terms of random Lie algebra elements and replaces the transience or Furstenberg’s irreducibility hypothesis in related prob- lems. The Birkhoff sum of any given smooth function then turns out to be equal to its integral w.r.t. a unique smooth measure on the manifold up to errors of the order of the coupling constant. Applica- tions to the theory of products of random matrices and a model of a disordered quantum wire are presented. 1. Main results, discussion and applications. This work provides a per- turbative calculation of invariant measures for a class of Markov chains on continuous state spaces and shows that these perturbative measures are unique and smooth. Let us state the main result right away in detail, and then place it into context with other work towards the end of this section and explain our motivation to study this problem. Suppose given a Lie group GL(L, C), a compact, connected, smooth Riemannian manifold withoutG⊂ boundary and a smooth, transitive group action : .M Thus, is a homogeneous space.
    [Show full text]
  • PETER-WEYL THEOREM and ITS APPLICATIONS 1. Some Functional
    LECTURE 23-24: PETER-WEYL THEOREM AND ITS APPLICATIONS 1. Some Functional Analysis Let H be a (complex) Hilbert space, i.e. a (finite or infinite dimensional) vector space with an inner product, such that H is complete with respect to the induced metric jvj = hv; vi1=2. A linear operator T : H ! H is said to be bounded if there exists C > 0 such that jT vj ≤ Cjvj; 8v 2 H: Definition 1.1. Let H be a Hilbert space, and T : H ! H a bounded operator. (1) T is self-adjoint if for any v; w 2 H, hT v; wi = hv; T wi. (2) T is compact if for any bounded sequence v1; v2; ··· in H, the sequence T v1; T v2; ··· has a convergent subsequence. We say that λ 2 C is an eigenvalue of an operator T on H, with eigenvector 0 6= v 2 H, if T v = λv. We will denote the set of all eigenvalues of T by Spec(T ), and the eigenspace for λ 2 Spec(T ) by Hλ. Self-adjoint operators on Hilbert spaces are the (infinite dimensional) generalization of n × n Hermitian matrices acting on Cn. For example, we have Lemma 1.2. If T is self-adjoint, then (1) Spec(T ) ⊂ R, (2) for any λ 6= µ 2 Spec(T ), Hλ ? Hµ. Proof. If λ 2 Spec(T ) with eigenvector v, and T is self-adjoint, then λhv; vi = hT v; vi = hv; T vi = hv; λvi = λ¯hv; vi: So λ = λ¯, i.e. λ 2 R.
    [Show full text]
  • Bounds Or Matrix Coefficients
    BOUNDS FOR MATRIX COEFFICIENTS AND ARITHMETIC APPLICATIONS by Ramin Takloo-Bighash Abstract. — We explain an important result of Hee Oh [16] on bounding matrix coefficients of semi-simple groups and survey some applications. Contents 1. Introduction .......................................................... 1 2. Uniform Pointwise Bounds for Matrix Coefficients .................... 4 3. Applications to equidistribution of Hecke points ...................... 6 4. Applications to rational points ........................................ 12 References ................................................................ 18 1. Introduction Let G be a connected reductive group over a local field F . We denote by G[(F ) the unitary dual of G(F ), that is the collection of equivalence classes of irreducible unitary representations of G(F ). G[(F ) has a natural topology known as the Fell Topology which is described as follows. We will introduce a basis of neighborhoods. Let ρ be any element of G[(F ), a positive number, φ1, . , φn diagonal matrix coefficients of ρ, and K a compact subset of G(F ). We define the open set W (φ1, . , φn,K; ; ρ) 0 0 to be the set of all η ∈ G[(F ) such that there exist φ1, . , φn each of which is a sum of diagonal matrix coefficients of η satisfying 0 |φi(x) − φi(x)| < The author was partially supported by a Young Investigator grant from the NSA. 2 RAMIN TAKLOO-BIGHASH for all x ∈ K and all i = 1, . , n. For more details see [9, 22]. We say G(F ) has Property T if the trivial representation is isolated in G[(F ). In concrete terms this means that if ρ is a non-trivial irreducible unitary representation of G(F ), and φ is a K-finite matrix coefficient of ρ, then φ has exponential decay in the sense that will be made explicit in the next section.
    [Show full text]
  • Spherical Functions on Spheres of Rank Two
    SPHERICAL FUNCTIONS ON SPHERES OF RANK TWO MAARTEN VAN PRUIJSSEN Abstract. Under a suitable multiplicity freeness assumption we show that the spherical functions of the pair (SU(n + 1); SU(n)) can be described by families of vector-valued poly- nomials in two variables. The matrix weight that establishes the orthogonality is supported on the closed unit disc and depends on n. We calculate the matrix weight in a couple of examples and we show that in these examples the weight depends actually on n 2 (1; 1). Contents 1. Introduction 1 2. Invariant functions 3 3. Multiplicity Free Induction 8 4. Vector-valued orthogonal polynomials 11 5. Examples 12 References 15 1. Introduction Consider the Gelfand pair (U; L) = (SU(n + 1); SU(n)) with n ≥ 2 and an irreducible representation πL : L ! GL(V ) with the following property: for every irreducible U- U 0 U L 0 representation π : U ! GL(V ) the restriction π jL contains π at most once, i.e. dim HomL(V; V ) ≤ 1. We are interested in the space of functions F : U ! End(V ) such that L L (1.1) F (`1u`2) = π (`1)F (u)π (`2) for all (`1; u; `2) 2 L × U × L: For example, if πL is the trivial representation, then the space of algebraic functions on U satisfying (1.1) can be given the structure of a polynomial algebra generated by two elements. The building blocks of this space are the zonal spherical functions, i.e. certain matrix coefficients of irreducible U-representations that contain an L-invariant vector.
    [Show full text]