Gelfand Pairs and Spherical Functions
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On the Stability of a Class of Functional Equations
Journal of Inequalities in Pure and Applied Mathematics ON THE STABILITY OF A CLASS OF FUNCTIONAL EQUATIONS volume 4, issue 5, article 104, BELAID BOUIKHALENE 2003. Département de Mathématiques et Informatique Received 20 July, 2003; Faculté des Sciences BP 133, accepted 24 October, 2003. 14000 Kénitra, Morocco. Communicated by: K. Nikodem EMail: [email protected] Abstract Contents JJ II J I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 Quit 098-03 Abstract In this paper, we study the Baker’s superstability for the following functional equation Z X −1 (E (K)) f(xkϕ(y)k )dωK (k) = |Φ|f(x)f(y), x, y ∈ G ϕ∈Φ K where G is a locally compact group, K is a compact subgroup of G, ωK is the normalized Haar measure of K, Φ is a finite group of K-invariant morphisms of On the Stability of A Class of G and f is a continuous complex-valued function on G satisfying the Kannap- Functional Equations pan type condition, for all x, y, z ∈ G Belaid Bouikhalene Z Z Z Z −1 −1 −1 −1 (*) f(zkxk hyh )dωK (k)dωK (h)= f(zkyk hxh )dωK (k)dωK (h). K K K K Title Page We treat examples and give some applications. Contents 2000 Mathematics Subject Classification: 39B72. Key words: Functional equation, Stability, Superstability, Central function, Gelfand JJ II pairs. The author would like to greatly thank the referee for his helpful comments and re- J I marks. Go Back Contents Close 1 Introduction, Notations and Preliminaries ............... -
Maximality of Hyperspecial Compact Subgroups Avoiding Bruhat–Tits Theory Tome 67, No 1 (2017), P
R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Marco MACULAN Maximality of hyperspecial compact subgroups avoiding Bruhat–Tits theory Tome 67, no 1 (2017), p. 1-21. <http://aif.cedram.org/item?id=AIF_2017__67_1_1_0> © Association des Annales de l’institut Fourier, 2017, Certains droits réservés. Cet article est mis à disposition selon les termes de la licence CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 3.0 FRANCE. http://creativecommons.org/licenses/by-nd/3.0/fr/ L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 67, 1 (2017) 1-21 MAXIMALITY OF HYPERSPECIAL COMPACT SUBGROUPS AVOIDING BRUHAT–TITS THEORY by Marco MACULAN Abstract. — Let k be a complete non-archimedean field (non trivially valued). Given a reductive k-group G, we prove that hyperspecial subgroups of G(k) (i.e. those arising from reductive models of G) are maximal among bounded subgroups. The originality resides in the argument: it is inspired by the case of GLn and avoids all considerations on the Bruhat–Tits building of G. Résumé. — Soit k un corps non-archimédien complet et non trivialement va- lué. Étant donné un k-groupe réductif G, nous démontrons que les sous-groupes hyperspéciaux de G(k) (c’est-à-dire ceux qui proviennent des modèles réductifs de G) sont maximaux parmi les sous-groupes bornés. -
Arxiv:1902.09497V5 [Math.DS]
GELFAND PAIRS ADMIT AN IWASAWA DECOMPOSITION NICOLAS MONOD Abstract. Every Gelfand pair (G,K) admits a decomposition G = KP , where P<G is an amenable subgroup. In particular, the Furstenberg boundary of G is homogeneous. Applications include the complete classification of non-positively curved Gelfand pairs, relying on earlier joint work with Caprace, as well as a canonical family of pure spherical functions in the sense of Gelfand–Godement for general Gelfand pairs. Let G be a locally compact group. The space M b(G) of bounded measures on G is an algebra for convolution, which is simply the push-forward of the multiplication map G × G → G. Definition. Let K<G be a compact subgroup. The pair (G,K) is a Gelfand pair if the algebra M b(G)K,K of bi-K-invariant measures is commutative. This definition, rooted in Gelfand’s 1950 work [14], is often given in terms of algebras of func- tions [12]. This is equivalent, by an approximation argument in the narrow topology, but has the inelegance of requiring the choice (and existence) of a Haar measure on G. Examples of Gelfand pairs include notably all connected semi-simple Lie groups G with finite center, where K is a maximal compact subgroup. Other examples are provided by their analogues over local fields [18], and non-linear examples include automorphism groups of trees [23],[2]. All these “classical” examples also have in common another very useful property: they admit a co-compact amenable subgroup P<G. In the semi-simple case, P is a minimal parabolic subgroup. -
The Classical Groups and Domains 1. the Disk, Upper Half-Plane, SL 2(R
(June 8, 2018) The Classical Groups and Domains Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ The complex unit disk D = fz 2 C : jzj < 1g has four families of generalizations to bounded open subsets in Cn with groups acting transitively upon them. Such domains, defined more precisely below, are bounded symmetric domains. First, we recall some standard facts about the unit disk, the upper half-plane, the ambient complex projective line, and corresponding groups acting by linear fractional (M¨obius)transformations. Happily, many of the higher- dimensional bounded symmetric domains behave in a manner that is a simple extension of this simplest case. 1. The disk, upper half-plane, SL2(R), and U(1; 1) 2. Classical groups over C and over R 3. The four families of self-adjoint cones 4. The four families of classical domains 5. Harish-Chandra's and Borel's realization of domains 1. The disk, upper half-plane, SL2(R), and U(1; 1) The group a b GL ( ) = f : a; b; c; d 2 ; ad − bc 6= 0g 2 C c d C acts on the extended complex plane C [ 1 by linear fractional transformations a b az + b (z) = c d cz + d with the traditional natural convention about arithmetic with 1. But we can be more precise, in a form helpful for higher-dimensional cases: introduce homogeneous coordinates for the complex projective line P1, by defining P1 to be a set of cosets u 1 = f : not both u; v are 0g= × = 2 − f0g = × P v C C C where C× acts by scalar multiplication. -
[Math.DG] 7 Feb 2007 Hoe 1.1
A GEOMETRIC PROOF OF THE KARPELEVICH-MOSTOW’S THEOREM ANTONIO J. DI SCALA AND CARLOS OLMOS Abstract. In this paper we give a geometric proof of the Karpelevich’s theo- rem that asserts that a semisimple Lie subgroup of isometries, of a symmetric space of non compact type, has a totally geodesic orbit. In fact, this is equiv- alent to a well-known result of Mostow about existence of compatible Cartan decompositions. 1. Introduction. In this paper we address the problem of giving a geometric proof of the following theorem of Karpelevich. Theorem 1.1. (Karpelevich [Kar53]) Let M be a Riemannian symmetric space of non positive curvature without flat factor. Then any connected and semisimple subgroup G ⊂ Iso(M) has a totally geodesic orbit G.p ⊂ M. It is well-known that Karpelevich’s theorem is equivalent to the following alge- braic theorem. Theorem 1.2. (Mostow [Mos55, Theorem 6]) Let g′ be a real semisimple Lie algebra of non compact type and let g ⊂ g′ be a semisimple Lie subalgebra. Let g = k⊕p be a Cartan decomposition for g. Then there exists a Cartan decomposition g′ = k′ ⊕ p′ for g′ such that k ⊂ k′ and p ⊂ p′. The proof of the above theorems is very algebraic in nature and uses delicate arguments related to automorphisms of semisimple Lie algebras (see [Oni04] for a modern exposition of Karpelevich’s results). For the real hyperbolic spaces, i.e. when g′ = so(n, 1), there are two geometric arXiv:math/0702201v1 [math.DG] 7 Feb 2007 proofs of Karpelevich’s theorem [DSO01], [BZ04]. -
MAT 449 : Representation Theory
MAT 449 : Representation theory Sophie Morel December 17, 2018 Contents I Representations of topological groups5 I.1 Topological groups . .5 I.2 Haar measures . .9 I.3 Representations . 17 I.3.1 Continuous representations . 17 I.3.2 Unitary representations . 20 I.3.3 Cyclic representations . 24 I.3.4 Schur’s lemma . 25 I.3.5 Finite-dimensional representations . 27 I.4 The convolution product and the group algebra . 28 I.4.1 Convolution on L1(G) and the group algebra of G ............ 28 I.4.2 Representations of G vs representations of L1(G) ............. 33 I.4.3 Convolution on other Lp spaces . 39 II Some Gelfand theory 43 II.1 Banach algebras . 43 II.1.1 Spectrum of an element . 43 II.1.2 The Gelfand-Mazur theorem . 47 II.2 Spectrum of a Banach algebra . 48 II.3 C∗-algebras and the Gelfand-Naimark theorem . 52 II.4 The spectral theorem . 55 III The Gelfand-Raikov theorem 59 III.1 L1(G) ....................................... 59 III.2 Functions of positive type . 59 III.3 Functions of positive type and irreducible representations . 65 III.4 The convex set P1 ................................. 68 III.5 The Gelfand-Raikov theorem . 73 IV The Peter-Weyl theorem 75 IV.1 Compact operators . 75 IV.2 Semisimplicity of unitary representations of compact groups . 77 IV.3 Matrix coefficients . 80 IV.4 The Peter-Weyl theorem . 86 IV.5 Characters . 87 3 Contents IV.6 The Fourier transform . 90 IV.7 Characters and Fourier transforms . 93 V Gelfand pairs 97 V.1 Invariant and bi-invariant functions . -
Arxiv:1810.00775V1 [Math-Ph] 1 Oct 2018 .Introduction 1
Modular structures and extended-modular-group-structures after Hecke pairs Orchidea Maria Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, Department of Theoretical Physics and Physics Education- KTFDF, Mlynsk´aDolina F2, 842 48, Bratislava, Slovakia; Sapienza University of Rome, Faculty of Civil and Industrial Engineering, DICEA- Department of Civil, Constructional and Environmental Engineering, Via Eudossiana, 18- 00184 Rome, Italy. E-mail: [email protected]; [email protected] Abstract. The simplices and the complexes arsing form the grading of the fundamental (desymmetrized) domain of arithmetical groups and non-arithmetical groups, as well as their extended (symmetrized) ones are described also for oriented manifolds in dim > 2. The conditions for the definition of fibers are summarized after Hamiltonian analysis, the latters can in some cases be reduced to those for sections for graded groups, such as the Picard groups and the Vinberg group.The cases for which modular structures rather than modular-group- structure measures can be analyzed for non-arithmetic groups, i.e. also in the cases for which Gelfand triples (rigged spaces) have to be substituted by Hecke couples, as, for Hecke groups, the existence of intertwining operators after the calculation of the second commutator within the Haar measures for the operators of the correspondingly-generated C∗ algebras is straightforward. The results hold also for (also non-abstract) groups with measures on (manifold) boundaries. The Poincar´e invariance of the representation of Wigner-Bargmann (spin 1/2) particles is analyzed within the Fock-space interaction representation. The well-posed-ness of initial conditions and boundary ones for the connected (families of) equations is discussed. -
Sato-Tate Equidistribution for Families of Hecke-Maass Forms on SL (N, R)/SO
SATO-TATE EQUIDISTRIBUTION FOR FAMILIES OF HECKE–MAASS FORMS ON SL(n, R)/ SO(n) JASMIN MATZ AND NICOLAS TEMPLIER Abstract. We establish the Sato-Tate equidistribution of Hecke eigenvalues of the family of Hecke–Maass cusp forms on SL(n, Z) SL(n, R)/ SO(n). As part of the proof, we establish a uniform upper-bound for spherical functions on\ semisimple Lie groups which is of independent interest. For each of the principal, symmetric square and exterior square L-functions, we deduce the level dis- tribution with restricted support of the low-lying zeros. We also deduce average estimates toward Ramanujan, including an improvement on the previous literature in the case n = 2. Contents 1. Introduction 1 2. Symmetry type of families and low-lying zeros 12 3. Average bounds towards Ramanujan 15 Part 1. Local theory: Real orbital integrals 18 4. Preliminaries 19 5. Setting for the main estimate 23 6. Bounds for semisimple orbital integrals 26 7. Weighted orbital integrals 29 8. Spherical functions and archimedean test functions 39 Part 2. Global theory: Weyl’s law and equidistribution 44 9. Setup for the Arthur-Selberg trace formula for GL(n) 44 10. Centralizers of semisimple elements 45 11. Coarse and fine geometric expansion 50 12. Bound for p-adic weighted orbital integrals 61 13. Spectral side and conclusion of proof of Theorem 1.1 67 References 72 arXiv:1505.07285v8 [math.NT] 18 May 2020 1. Introduction Hecke–Maass cusp forms are certain eigenfunctions of the Laplace operator on the locally sym- metric space SL(n, Z) SL(n, R)/ SO(n). -
Homogeneous Spaces with the Cohomology of Sphere Products and Compact Quadrangles
Homogeneous spaces with the cohomology of sphere products and compact quadrangles Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der Bayerischen Julius-Maximilians-Universit¨at Wurzburg¨ vorgelegt von Oliver Bletz-Siebert aus Heilbronn Wurzburg¨ 2002 Contents Preface v Notation xi 1 Fibrations and double fibrations 1 1.1 Fibrations . 1 1.2 Double fibrations . 2 2 Lie group actions 11 2.1 Lie theory . 12 2.2 Group actions . 18 2.3 Transitive actions . 21 2.4 Homogeneous spheres . 22 2.5 Almost transitive actions . 26 3 Isoparametric hypersurfaces 33 3.1 Homogeneous hypersurfaces . 33 3.2 Transitive actions on focal manifolds . 35 4 Generalized quadrangles 39 4.1 Geometries and generalized quadrangles . 39 4.2 Homogeneous quadrangles . 42 4.3 Point-homogeneous (1; 2)-quadrangles . 47 4.4 Orthogonal actions . 49 iii iv CONTENTS 4.5 Unitary actions . 51 4.6 Summary . 53 5 Three series of homogeneous spaces 55 5.1 The (5; 4n − 6)-series . 55 5.2 The (7; 4n − 8)-series . 58 5.3 The (3; 2n − 2)-series . 59 6 Rational cohomology 61 6.1 Orientable fibrations . 62 6.2 Spectral sequences . 68 6.3 Rational homotopy and rational cohomology . 74 6.4 Cohomology of some homogeneous spaces . 79 Bibliography 89 Index 94 Preface The homogeneous spaces that have the same (singular) cohomology as spheres were classified by Borel, Bredon, Montgomery and Samelson and by Poncet. This was extended to homogeneous spaces which are simply connected and which have the same rational cohomology as spheres by Onishchik and also by Kramer. Furthermore, Kramer classified the simply connected homoge- neous spaces with the rational cohomology of a sphere product k × m, where 3 ≤ k ≤ m and m is odd; and Wolfrom classified this kind of spaces in the case 2 = k ≤ m with odd m. -
Commuting Tuples in Reductive Groups and Their Maximal Compact Subgroups
Geometry & Topology 17 (2013) 2513–2593 msp Commuting tuples in reductive groups and their maximal compact subgroups ALEXANDRA PETTET JUAN SOUTO Let G be a reductive algebraic group and K G a maximal compact subgroup. We consider the representation spaces Hom.Zk ; K/ and Hom.Zk ; G/ with the topology induced from an embedding into Kk and Gk , respectively. The goal of this paper is to prove that Hom.Zk ; K/ is a strong deformation retract of Hom.Zk ; G/. 20G20; 55P99 1 Introduction k By a commuting k –tuple in a group G we mean a k –tuple .a1;:::; a / G with k 2 aiaj aj ai for all i and j . Observe that any such commuting k –tuple determines D uniquely a homomorphism k n1 nk Z G;.n1;:::; n / a a : ! k 7! 1 k In fact, we obtain in this way a bijection between Hom.Zk ; G/ and the set of commuting k –tuples in G . Suppose now that G is a Lie group and that K G is a closed subgroup. Endowing Hom.Zk ; G/ Gk and Hom.Zk ; K/ Kk with the subspace topology, we consider the embedding (1.1) Hom.Zk ; K/, Hom.Zk ; G/ ! induced by the inclusion of K into G . The goal of this paper is to prove that the map (1.1) is a homotopy equivalence if G is a reductive algebraic group and K G is a maximal compact subgroup. Before stating our main result precisely, recall that a weak retraction of a topological space X to a subspace A is a continuous map F Œ0; 1 X X;.t; x/ Ft .x/ W ! 7! Per me si va ne la cittá dolente, per me si va ne l’etterno dolore, per me si va tra la perduta gente. -
Multiplicity One Theorem for GL(N) and Other Gelfand Pairs
Multiplicity one theorem for GL(n) and other Gelfand pairs A. Aizenbud Massachusetts Institute of Technology http://math.mit.edu/~aizenr A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs L2(S1) = L spanfeinx g geinx = χ(g)einx Example (Spherical Harmonics) 2 2 L m L (S ) = m H m m H = spani fyi g are irreducible representations of O3 Example Let X be a finite set. Let the symmetric group Perm(X) act on X. Consider the space F(X) of complex valued functions on X as a representation of Perm(X). Then it decomposes to direct sum of distinct irreducible representations. Examples Example (Fourier Series) A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs geinx = χ(g)einx Example (Spherical Harmonics) 2 2 L m L (S ) = m H m m H = spani fyi g are irreducible representations of O3 Example Let X be a finite set. Let the symmetric group Perm(X) act on X. Consider the space F(X) of complex valued functions on X as a representation of Perm(X). Then it decomposes to direct sum of distinct irreducible representations. Examples Example (Fourier Series) L2(S1) = L spanfeinx g A. Aizenbud Multiplicity one theorem for GL(n) and other Gelfand pairs Example (Spherical Harmonics) 2 2 L m L (S ) = m H m m H = spani fyi g are irreducible representations of O3 Example Let X be a finite set. Let the symmetric group Perm(X) act on X. Consider the space F(X) of complex valued functions on X as a representation of Perm(X). -
The Action of a Semisimple Lie Group on Its Maximal Compact Subgroup
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 5, Pages 1525{1534 S 0002-9939(01)05984-6 Article electronically published on January 8, 2001 THE ACTION OF A SEMISIMPLE LIE GROUP ON ITS MAXIMAL COMPACT SUBGROUP T. BUDAK, N. IS¸IK, P. MILNES, AND J. PYM (Communicated by Michael Handel) Abstract. In this paper we determine the structure of the minimal ideal in the enveloping semigroup for the natural action of a connected semisimple Lie group on its maximal compact subgroup. In particular, if G = KAN is an Iwasawa decomposition of the group G, then the group in the minimal left ideal is isomorphic both algebraically and topologically with the normalizer M of AN in K: Complete descriptions are given for the enveloping semigroups in the cases G =SL(2; C)andG =SL(2; R): 1. Introduction Throughout this paper G will be a connected semisimple Lie group with finite centre. It therefore has an Iwasawa decomposition, that is G = KAN is a product of closed connected subgroups with K compact, A abelian and N nilpotent, and the decomposition g = kan of any element of G is unique. The prime examples of these groups are the special linear groups SL(n; R) and SL(n; C); here a suit- able decomposition can be obtained by taking K =SO(n; R) in the real case and SU(n) in the complex case, A to be the diagonal matrices with positive entries and determinant 1, and N to be the upper triangular matrices with 1's on the diagonal. The subgroup AN is closed and the homogeneous space G=AN is naturally homeomorphic to the compact space K: G therefore has a natural action on K;if we identify the coset gAN with k 2 K we can indicate the action of g0 2 G by g0 : k = gAN 7! g0(k)=g0gAN: The restriction of this action to K ⊂ G is just left multiplication by members of K; that is, k0(k)=k0k: Thus these flows (G; K) are not only minimal (each orbit is dense) but transitive (given k; k0 2 K we can find g 2 G with gk = k0).