Geometry of Character Varieties of Surface Groups
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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. -
Chapter 2 C -Algebras
Chapter 2 C∗-algebras This chapter is mainly based on the first chapters of the book [Mur90]. Material bor- rowed from other references will be specified. 2.1 Banach algebras Definition 2.1.1. A Banach algebra C is a complex vector space endowed with an associative multiplication and with a norm k · k which satisfy for any A; B; C 2 C and α 2 C (i) (αA)B = α(AB) = A(αB), (ii) A(B + C) = AB + AC and (A + B)C = AC + BC, (iii) kABk ≤ kAkkBk (submultiplicativity) (iv) C is complete with the norm k · k. One says that C is abelian or commutative if AB = BA for all A; B 2 C . One also says that C is unital if 1 2 C , i.e. if there exists an element 1 2 C with k1k = 1 such that 1B = B = B1 for all B 2 C . A subalgebra J of C is a vector subspace which is stable for the multiplication. If J is norm closed, it is a Banach algebra in itself. Examples 2.1.2. (i) C, Mn(C), B(H), K (H) are Banach algebras, where Mn(C) denotes the set of n × n-matrices over C. All except K (H) are unital, and K (H) is unital if H is finite dimensional. (ii) If Ω is a locally compact topological space, C0(Ω) and Cb(Ω) are abelian Banach algebras, where Cb(Ω) denotes the set of all bounded and continuous complex func- tions from Ω to C, and C0(Ω) denotes the subset of Cb(Ω) of functions f which vanish at infinity, i.e. -
Notes on the Chern-Character Maakestad H* Department of Mathematics, NTNU, Trondheim, Norway
Theory an ie d L A p d p Maakestad, J Generalized Lie Theory Appl 2017, 11:1 e l z i i c l a a t Journal of Generalized Lie r DOI: 10.4172/1736-4337.1000253 i o e n n s e G ISSN: 1736-4337 Theory and Applications Research Article Open Access Notes on the Chern-Character Maakestad H* Department of Mathematics, NTNU, Trondheim, Norway Abstract Notes for some talks given at the seminar on characteristic classes at NTNU in autumn 2006. In the note a proof of the existence of a Chern-character from complex K-theory to any cohomology Lie theory with values in graded Q-algebras equipped with a theory of characteristic classes is given. It respects the Adams and Steenrod operations. Keywords: Chern-character; Chern-classes; Euler classes; Singular H* : Top→−Q algebras cohomology; De Rham-cohomology; Complex K-theory; Adams operations; Steenrod operations from the category of topological spaces to the category of graded commutative Q-algebras with respect to continuous maps of topological Introduction spaces. We say the theory satisfy the projective bundle property if the The aim of this note is to give an axiomatic and elementary following axioms are satisfied: For any rankn complex continuous treatment of Chern-characters of vectorbundles with values in a vectorbundle E over a compact space B There is an Euler class. class of cohomology-theories arising in topology and algebra. Given ∈ 2 uE H (P(E)) (1) a theory of Chern-classes for complex vectorbundles with values in → singular cohomology one gets in a natural way a Chern-character from Where π:P(E) B is the projective bundle associated to E. -
GROUP REPRESENTATIONS and CHARACTER THEORY Contents 1
GROUP REPRESENTATIONS AND CHARACTER THEORY DAVID KANG Abstract. In this paper, we provide an introduction to the representation theory of finite groups. We begin by defining representations, G-linear maps, and other essential concepts before moving quickly towards initial results on irreducibility and Schur's Lemma. We then consider characters, class func- tions, and show that the character of a representation uniquely determines it up to isomorphism. Orthogonality relations are introduced shortly afterwards. Finally, we construct the character tables for a few familiar groups. Contents 1. Introduction 1 2. Preliminaries 1 3. Group Representations 2 4. Maschke's Theorem and Complete Reducibility 4 5. Schur's Lemma and Decomposition 5 6. Character Theory 7 7. Character Tables for S4 and Z3 12 Acknowledgments 13 References 14 1. Introduction The primary motivation for the study of group representations is to simplify the study of groups. Representation theory offers a powerful approach to the study of groups because it reduces many group theoretic problems to basic linear algebra calculations. To this end, we assume that the reader is already quite familiar with linear algebra and has had some exposure to group theory. With this said, we begin with a preliminary section on group theory. 2. Preliminaries Definition 2.1. A group is a set G with a binary operation satisfying (1) 8 g; h; i 2 G; (gh)i = g(hi)(associativity) (2) 9 1 2 G such that 1g = g1 = g; 8g 2 G (identity) (3) 8 g 2 G; 9 g−1 such that gg−1 = g−1g = 1 (inverses) Definition 2.2. -
Geometric Structures and Varieties of Representations
GEOMETRIC STRUCTURES AND VARIETIES OF REPRESENTATIONS WILLIAM M. GOLDMAN Abstract. Many interesting geometric structures on manifolds can be interpreted as structures locally modelled on homogeneous spaces. Given a homogeneous space (X; G) and a manifold M, there is a deformation space of structures on M locally modelled on the geometry of X invariant under G. Such a geometric struc- ture on a manifold M determines a representation (unique up to inner automorphism) of the fundamental group ¼ of M in G. The deformation space for such structures is \locally modelled" on the space Hom(¼; G)=G of equivalence classes of representations of ¼ ! G. A strong interplay exists between the local and global structure of the variety of representations and the corresponding geometric structures. The lecture in Boulder surveyed some as- pects of this correspondence, focusing on: (1) the \Deformation Theorem" relating deformation spaces of geometric structures to the space of representations; (2) representations of surface groups in SL(2; R), hyperbolic structures on surfaces (with singularities), Fenchel-Nielsen coordinates on TeichmullerÄ space; (3) convex real projective structures on surfaces; (4) representations of Schwarz triangle groups in SL(3; C). This paper represents an expanded version of the lecture. Introduction According to Felix Klein's Erlanger program of 1872, geometry is the study of the properties of a space which are invariant under a group of transformations. A geometry in Klein's sense is thus a pair (X; G) where X is a manifold and G is a Lie group acting (transitively) on X. If (X; G) is Euclidean geometry (so X = Rn and G is its group of isometries), then one might \in¯nitesimally model" a manifold M on Euclidean space by giving each tangent space a Euclidean geometry, which varies from point to point; the resulting in¯nitesimally Euclidean geometry is a Riemannian metric. -
Conditions of Smoothness of Moduli Spaces of Flat Connections and of Character Varieties
This is a repository copy of Conditions of smoothness of moduli spaces of flat connections and of character varieties. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/149815/ Version: Accepted Version Article: Ho, Nankuo, Wilkin, Graeme Peter Desmond and Wu, Siye (2018) Conditions of smoothness of moduli spaces of flat connections and of character varieties. Mathematische Zeitschrift. ISSN 1432-1823 https://doi.org/10.1007/s00209-018-2158-2 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Conditions of smoothness of moduli spaces of flat connections and of character varieties Nan-Kuo Ho1,a, Graeme Wilkin2,b and Siye Wu1,c 1 Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan 2 Department of Mathematics, National University of Singapore, Singapore 119076 a E-mail address: [email protected] b E-mail address: [email protected] c E-mail address: [email protected] Abstract. -
Representation Theory As Gauge Theory
Representation theory Quantum Field Theory Gauge Theory Representation Theory as Gauge Theory David Ben-Zvi University of Texas at Austin Clay Research Conference Oxford, September 2016 Representation theory Quantum Field Theory Gauge Theory Themes I. Harmonic analysis as the exploitation of symmetry1 II. Commutative algebra signals geometry III. Topology provides commutativity IV. Gauge theory bridges topology and representation theory 1Mackey, Bull. AMS 1980 Representation theory Quantum Field Theory Gauge Theory Themes I. Harmonic analysis as the exploitation of symmetry1 II. Commutative algebra signals geometry III. Topology provides commutativity IV. Gauge theory bridges topology and representation theory 1Mackey, Bull. AMS 1980 Representation theory Quantum Field Theory Gauge Theory Themes I. Harmonic analysis as the exploitation of symmetry1 II. Commutative algebra signals geometry III. Topology provides commutativity IV. Gauge theory bridges topology and representation theory 1Mackey, Bull. AMS 1980 Representation theory Quantum Field Theory Gauge Theory Themes I. Harmonic analysis as the exploitation of symmetry1 II. Commutative algebra signals geometry III. Topology provides commutativity IV. Gauge theory bridges topology and representation theory 1Mackey, Bull. AMS 1980 Representation theory Quantum Field Theory Gauge Theory Outline Representation theory Quantum Field Theory Gauge Theory Representation theory Quantum Field Theory Gauge Theory Fourier Series G = U(1) acts on S1, hence on L2(S1): Fourier series 2 1 [M 2πinθ L (S ) ' Ce n2Z joint eigenspaces of rotation operators See last slide for all image credits Representation theory Quantum Field Theory Gauge Theory The dual Basic object in representation theory: describe the dual of a group G: Gb = f irreducible (unitary) representations of Gg e.g. -
L-Functions and Non-Abelian Class Field Theory, from Artin to Langlands
L-functions and non-abelian class field theory, from Artin to Langlands James W. Cogdell∗ Introduction Emil Artin spent the first 15 years of his career in Hamburg. Andr´eWeil charac- terized this period of Artin's career as a \love affair with the zeta function" [77]. Claude Chevalley, in his obituary of Artin [14], pointed out that Artin's use of zeta functions was to discover exact algebraic facts as opposed to estimates or approxi- mate evaluations. In particular, it seems clear to me that during this period Artin was quite interested in using the Artin L-functions as a tool for finding a non- abelian class field theory, expressed as the desire to extend results from relative abelian extensions to general extensions of number fields. Artin introduced his L-functions attached to characters of the Galois group in 1923 in hopes of developing a non-abelian class field theory. Instead, through them he was led to formulate and prove the Artin Reciprocity Law - the crowning achievement of abelian class field theory. But Artin never lost interest in pursuing a non-abelian class field theory. At the Princeton University Bicentennial Conference on the Problems of Mathematics held in 1946 \Artin stated that `My own belief is that we know it already, though no one will believe me { that whatever can be said about non-Abelian class field theory follows from what we know now, since it depends on the behavior of the broad field over the intermediate fields { and there are sufficiently many Abelian cases.' The critical thing is learning how to pass from a prime in an intermediate field to a prime in a large field. -
Topological Quantum Field Theory for Character Varieties
Universidad Complutense de Madrid Topological Quantum Field Theories for Character Varieties Teor´ıasTopol´ogicasde Campos Cu´anticos para Variedades de Caracteres Angel´ Gonzalez´ Prieto Directores Prof. Marina Logares Jim´enez Prof. Vicente Mu~nozVel´azquez Tesis presentada para optar al grado de Doctor en Investigaci´onMatem´atica Departamento de Algebra,´ Geometr´ıay Topolog´ıa Facultad de Ciencias Matem´aticas Madrid, 2018 Universidad Complutense de Madrid Topological Quantum Field Theories for Character Varieties Angel´ Gonzalez´ Prieto Advisors: Prof. Marina Logares and Prof. Vicente Mu~noz Thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematical Research Departamento de Algebra,´ Geometr´ıay Topolog´ıa Facultad de Ciencias Matem´aticas Madrid, 2018 \It's not who I am underneath, but what I do that defines me." Batman ABSTRACT Topological Quantum Field Theory for Character Varieties by Angel´ Gonzalez´ Prieto Abstract en Espa~nol La presente tesis doctoral est´adedicada al estudio de las estructuras de Hodge de un tipo especial de variedades algebraicas complejas que reciben el nombre de variedades de caracteres. Para este fin, proponemos utilizar una poderosa herramienta de naturaleza ´algebro-geom´etrica,proveniente de la f´ısicate´orica,conocida como Teor´ıaTopol´ogicade Campos Cu´anticos (TQFT por sus siglas en ingl´es). Con esta idea, en la presente tesis desarrollamos un formalismo que nos permite construir TQFTs a partir de dos piezas m´assencillas de informaci´on: una teor´ıade campos (informaci´ongeom´etrica)y una cuantizaci´on(informaci´onalgebraica). Como aplicaci´on,construimos una TQFT que calcula las estructuras de Hodge de variedades de representaciones y la usamos para calcular expl´ıcitamente los polinomios de Deligne-Hodge de SL2(C)-variedades de caracteres parab´olicas. -
Character Varieties in SL 2 and Kauffman Skein Algebras
CHARACTER VARIETIES IN SL2 AND KAUFFMAN SKEIN ALGEBRAS JULIEN MARCHE´ Abstract. These lecture notes concern the algebraic geometry of the character variety of a finitely generated group in SL2(C) from the point of view of skein modules. We focus on the case of surface and 3-manifolds groups and construct the Reidemeister torsion as a rational volume form on the character variety. 1. Introduction These notes collect some general facts about character varieties of finitely generated groups in SL2. We stress that one can study character varieties with the point of view of skein modules: using a theorem of K. Saito, we recover some standard results of character varieties, including a construction of a so-called tautological representation. This allows to give a global construction of the Reidemeister torsion, which should be useful for further study such as its singularities or differential equations it should satisfy. The main motivation of the author is to understand the relation between character varieties and topological quantum field theory (TQFT) with gauge group SU2. This theory - in the [BHMV95] version - makes fundamental use of the Kauffman bracket skein module, an object intimately related to character varieties. Moreover the non-abelian Reidemeister torsion plays a fundamental role in the Witten asymptotic expansion conjecture, governing the asymptotics of quantum invariants of 3-manifolds. However, these notes do not deal with TQFT, and strictly speaking do not contain any new result. Let us describe and comment its content. (1) The traditional definition of character varieties uses an algebraic quotient, although it is not always presented that way. -
Varieties of Characters
VARIETIES OF CHARACTERS SEAN LAWTON AND ADAM S. SIKORA Abstract. Let G be a connected reductive affine algebraic group. In this short note we define the variety of G-characters of a finitely gener- ated group Γ and show that the quotient of the G-character variety of Γ by the action of the trace preserving outer automorphisms of G nor- malizes the variety of G-characters when Γ is a free group, free abelian group, or a surface group. 1. Introduction For a finitely generated group Γ and a connected reductive affine algebraic group G over the complex numbers, the G-character variety of Γ is the Geometric Invariant Theory (GIT) quotient X (Γ, G) := Hom(Γ, G)//G of the G-representation space of Γ, Hom(Γ, G), (considered as an algebraic set) by the adjoint action of G. This algebraic set is known for its relationship to moduli spaces of geo- metric structures ([CG97, Gol88, GM88, JM87, KM98, Thu97]), 3 and 4 dimensional topology ([CCG+94, CS83, BZ98, PS00, Bul97, Cur01]), mathe- matical physics ([Ati90, AB83, BFM02, Hit87, KS00, Wit01, JW92, KW07]) and invariant theory ([BH95, Law08, Sik01]). The term “G-character variety” is motivated by the fact that its closed points for G = SL(n, C), Sp(n, C), SO(2n + 1, C) are classified by the G- characters of Γ. (Perhaps the first proof of this was given by Culler-Shalen in [CS83] for the case SL(2, C).) However, it is not the case for SO(2n, C), see [FL11, Appendix A], [Sik15, Sik17]. -
Representation Theory
Representation Theory Drewtheph 1 Group ring and group algebra 1.1 F -algebra If F is a field, then an algebra over F (or simply an F -algebra) is a set A with a ring structure and an F -vector space structure that share the same addition operation, and with the additional property that (Aa)b = A(ab) = a(Ab) for any λ 2 F and a; b 2 A. 1.2 Semisimple A module is said to be semisimple if it is a direct sum of simple modules. We shall say that the algebra A is semisimple if all non-zero A−modules are semisimple 1.3 Group ring Let K be a ring and let G be a multiplicative group, not necessarily finite. Then the group ring K[G] is a K−vector space with basis G 1.4 Group algebra FG The group algebra of G over F (F -algebra), denoted by FG, consists of all finite formal F -linear combinations of elements of F , i.e. a F -vector space with basis G, hence with dimension jGj. CG is a C-algebra 1.5 Dimension of CG-modules CG as a C-algebra and M an CG-module. Then M is also a C-vector space, where defining scalar multiplication by: λ 2 C; m 2 M, λm := (λ1CG):m, and addition is from the module structure. And the dimension of a CG-module M, is its dimension as a C-vector space. 1 1.6 One dimensional FG-module is simple If S is a nontrivial submodule of one dimensional FG-module M, since S is also a F -vector space, its dimension and be 0 or 1.