Reductive Subgroups of Reductive Groups in Nonzero Characteristic
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Automorphism Groups of Locally Compact Reductive Groups
PROCEEDINGS OF THE AMERICAN MATHEMATICALSOCIETY Volume 106, Number 2, June 1989 AUTOMORPHISM GROUPS OF LOCALLY COMPACT REDUCTIVE GROUPS T. S. WU (Communicated by David G Ebin) Dedicated to Mr. Chu. Ming-Lun on his seventieth birthday Abstract. A topological group G is reductive if every continuous finite di- mensional G-module is semi-simple. We study the structure of those locally compact reductive groups which are the extension of their identity components by compact groups. We then study the automorphism groups of such groups in connection with the groups of inner automorphisms. Proposition. Let G be a locally compact reductive group such that G/Gq is compact. Then /(Go) is dense in Aq(G) . Let G be a locally compact topological group. Let A(G) be the group of all bi-continuous automorphisms of G. Then A(G) has the natural topology (the so-called Birkhoff topology or ^-topology [1, 3, 4]), so that it becomes a topo- logical group. We shall always adopt such topology in the following discussion. When G is compact, it is well known that the identity component A0(G) of A(G) is the group of all inner automorphisms induced by elements from the identity component G0 of G, i.e., A0(G) = I(G0). This fact is very useful in the study of the structure of locally compact groups. On the other hand, it is also well known that when G is a semi-simple Lie group with finitely many connected components A0(G) - I(G0). The latter fact had been generalized to more general groups ([3]). -
LIE GROUPS and ALGEBRAS NOTES Contents 1. Definitions 2
LIE GROUPS AND ALGEBRAS NOTES STANISLAV ATANASOV Contents 1. Definitions 2 1.1. Root systems, Weyl groups and Weyl chambers3 1.2. Cartan matrices and Dynkin diagrams4 1.3. Weights 5 1.4. Lie group and Lie algebra correspondence5 2. Basic results about Lie algebras7 2.1. General 7 2.2. Root system 7 2.3. Classification of semisimple Lie algebras8 3. Highest weight modules9 3.1. Universal enveloping algebra9 3.2. Weights and maximal vectors9 4. Compact Lie groups 10 4.1. Peter-Weyl theorem 10 4.2. Maximal tori 11 4.3. Symmetric spaces 11 4.4. Compact Lie algebras 12 4.5. Weyl's theorem 12 5. Semisimple Lie groups 13 5.1. Semisimple Lie algebras 13 5.2. Parabolic subalgebras. 14 5.3. Semisimple Lie groups 14 6. Reductive Lie groups 16 6.1. Reductive Lie algebras 16 6.2. Definition of reductive Lie group 16 6.3. Decompositions 18 6.4. The structure of M = ZK (a0) 18 6.5. Parabolic Subgroups 19 7. Functional analysis on Lie groups 21 7.1. Decomposition of the Haar measure 21 7.2. Reductive groups and parabolic subgroups 21 7.3. Weyl integration formula 22 8. Linear algebraic groups and their representation theory 23 8.1. Linear algebraic groups 23 8.2. Reductive and semisimple groups 24 8.3. Parabolic and Borel subgroups 25 8.4. Decompositions 27 Date: October, 2018. These notes compile results from multiple sources, mostly [1,2]. All mistakes are mine. 1 2 STANISLAV ATANASOV 1. Definitions Let g be a Lie algebra over algebraically closed field F of characteristic 0. -
1:34 Pm April 11, 2013 Red.Tex
1:34 p.m. April 11, 2013 Red.tex The structure of reductive groups Bill Casselman University of British Columbia, Vancouver [email protected] An algebraic group defined over F is an algebraic variety G with group operations specified in algebraic terms. For example, the group GLn is the subvariety of (n + 1) × (n + 1) matrices A 0 0 a with determinant det(A) a = 1. The matrix entries are well behaved functions on the group, here for 1 example a = det− (A). The formulas for matrix multiplication are certainly algebraic, and the inverse of a matrix A is its transpose adjoint times the inverse of its determinant, which are both algebraic. Formally, this means that we are given (a) an F •rational multiplication map G × G −→ G; (b) an F •rational inverse map G −→ G; (c) an identity element—i.e. an F •rational point of G. I’ll look only at affine algebraic groups (as opposed, say, to elliptic curves, which are projective varieties). In this case, the variety G is completely characterized by its affine ring AF [G], and the data above are respectively equivalent to the specification of (a’) an F •homomorphism AF [G] −→ AF [G] ⊗F AF [G]; (b’) an F •involution AF [G] −→ AF [G]; (c’) a distinguished homomorphism AF [G] −→ F . The first map expresses a coordinate in the product in terms of the coordinates of its terms. For example, in the case of GLn it takes xik −→ xij ⊗ xjk . j In addition, these data are subject to the group axioms. I’ll not say anything about the general theory of such groups, but I should say that in practice the specification of an algebraic group is often indirect—as a subgroup or quotient, say, of another simpler one. -
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. -
Linear Algebraic Groups
Clay Mathematics Proceedings Volume 4, 2005 Linear Algebraic Groups Fiona Murnaghan Abstract. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups. 1. Algebraic groups Let K be an algebraically closed field. An algebraic K-group G is an algebraic variety over K, and a group, such that the maps µ : G × G → G, µ(x, y) = xy, and ι : G → G, ι(x)= x−1, are morphisms of algebraic varieties. For convenience, in these notes, we will fix K and refer to an algebraic K-group as an algebraic group. If the variety G is affine, that is, G is an algebraic set (a Zariski-closed set) in Kn for some natural number n, we say that G is a linear algebraic group. If G and G′ are algebraic groups, a map ϕ : G → G′ is a homomorphism of algebraic groups if ϕ is a morphism of varieties and a group homomorphism. Similarly, ϕ is an isomorphism of algebraic groups if ϕ is an isomorphism of varieties and a group isomorphism. A closed subgroup of an algebraic group is an algebraic group. If H is a closed subgroup of a linear algebraic group G, then G/H can be made into a quasi- projective variety (a variety which is a locally closed subset of some projective space). If H is normal in G, then G/H (with the usual group structure) is a linear algebraic group. Let ϕ : G → G′ be a homomorphism of algebraic groups. Then the kernel of ϕ is a closed subgroup of G and the image of ϕ is a closed subgroup of G. -
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. -
Geometry of Character Varieties of Surface Groups
GEOMETRY OF CHARACTER VARIETIES OF SURFACE GROUPS MOTOHICO MULASE∗ Abstract. This article is based on a talk delivered at the RIMS{OCAMI Joint International Conference on Geometry Related to Integrable Systems in September, 2007. Its aim is to review a recent progress in the Hitchin integrable systems and character varieties of the fundamental groups of Riemann surfaces. A survey on geometric aspects of these character varieties is also provided as we develop the exposition from a simple case to more elaborate cases. Contents 1. Introduction 1 2. Character varieties of finite groups and representation theory 2 3. Character varieties of Un as moduli spaces of stable vector bundles 5 4. Twisted character varieties of Un 6 5. Twisted character varieties of GLn(C) 10 6. Hitchin integrable systems 13 7. Symplectic quotient of the Hitchin system and mirror symmetry 16 References 21 1. Introduction The character varieties we consider in this article are the set of equivalence classes Hom(π1(Σg);G)=G of representations of a surface group π1(Σg) into another group G. Here Σg is a closed oriented surface of genus g, which is assumed to be g ≥ 2 most of the time. The action of G on the space of homomorphisms is through the conjugation action. Since this action has fixed points, the quotient requires a special treatment to make it a reasonable space. Despite the simple appearance of the space, it has an essential connection to many other subjects in mathematics ([1, 2, 6, 9, 10, 11, 14, 15, 17, 19, 24, 25, 33, 34]), and the list is steadily growing ([4, 7, 12, 13, 18, 23]). -
Real Group Orbits on Flag Manifolds
Real group orbits on flag manifolds Dmitri Akhiezer Abstract In this survey, we gather together various results on the action of a real form G0 of a complex semisimple group G on its flag manifolds. We start with the finiteness theorem of J. Wolf implying that at least one of the G0-orbits is open. We give a new proof of the converse statement for real forms of inner type, essentially due to F.M. Malyshev. Namely, if a real form of inner type G0 ⊂ G has an open orbit on a complex algebraic homogeneous space G/H then H is parabolic. In order to prove this, we recall, partly with proofs, some results of A.L. Onishchik on the factorizations of reductive groups. Finally, we discuss the cycle spaces of open G0-orbits and define the crown of a symmetric space of non- compact type. With some exceptions, the cycle space agrees with the crown. We sketch a complex analytic proof of this result, due to G. Fels, A. Huckleberry and J. Wolf. 1 Introduction The first systematic treatment of the orbit structure of a complex flag manifold X = G/P under the action of a real form G0 ⊂ G is due to J. Wolf [38]. Forty years after his paper, these real group orbits and their cycle spaces are still an object of intensive research. We present here some results in this area, together with other related results on transitive and locally transitive actions of Lie groups on complex manifolds. The paper is organized as follows. In Section 2 we prove the celebrated finiteness theorem for G0-orbits on X (Theorem 2.3). -
NON-SPLIT REDUCTIVE GROUPS OVER Z Brian Conrad
NON-SPLIT REDUCTIVE GROUPS OVER Z Brian Conrad Abstract. | We study the following phenomenon: some non-split connected semisimple Q-groups G admit flat affine Z-group models G with \everywhere good reduction" (i.e., GFp is a connected semisimple Fp-group for every prime p). Moreover, considering such G up to Z-group isomorphism, there can be more than one such G for a given G. This is seen classically for types B and D by using positive-definite quadratic lattices. The study of such Z-groups provides concrete applications of many facets of the theory of reductive groups over rings (scheme of Borel subgroups, auto- morphism scheme, relative non-abelian cohomology, etc.), and it highlights the role of number theory (class field theory, mass formulas, strong approximation, point-counting over finite fields, etc.) in analyzing the possibilities. In part, this is an expository account of [G96]. R´esum´e. | Nous ´etudions le ph´enom`ene suivant : certains Q-groupes G semi-simples connexes non d´eploy´es admettent comme mod`eles des Z-groupes G affines et plats avec \partout bonne r´eduction" (c'est `adire, GFp est un Fp- groupe Q-groupes G pour chaque premier p). En outre, consid´erant de tels G `a Z-groupe isomorphisme pr`es, il y a au plus un tel G pour un G donn´e.Ceci est vu classiquement pour les types B et D en utilisant des r´eseaux quadratiques d´efinis positifs. L'´etude de ces Z-groupes donne lieu `ades applications concr`etes d'aspects multiples, de la th´eorie des groupes r´eductifs sur des anneaux (sch´emas de sous-groupes de Borel, sch´emas d'automorphismes, cohomologie relative non ab´elienne, etc.), et met en ´evidence le r^ole de la th´eorie des nombres (th´eorie du corps de classes, formules de masse, approximation forte, comptage de points sur les corps finis, etc.) dans l'analyse des possibilit´es.En partie, ceci est un article d'exposition sur [G96]. -
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.