DOCSLIB.ORG

Buildings and Their Applications in Geometry and Topology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Buildings and Their Applications in Geometry and Topology Buildings and their applications in geometry and topology Lizhen Ji∗ September 21, 2011 Abstract In this paper, we briefly introduce different types of buildings such as spherical buildings, Euclidean buildings, twin buildings, R-buildings and de- scribe some of their applications to many subjects: (1) differential geom- etry such as Mostow strong rigidity, rank rigidity of nonpositively curved manifolds, Margulis superrigidity, quasi-isometry rigidity, classification of isoparametric manifolds, compactifications of symmetric spaces and locally symmetric spaces, (2) topology such as cohomology and duality properties of arithmetic groups, simplicial volume of locally symmetric spaces and Novikov conjectures, (3) analysis such as harmonic maps, harmonic analysis and rep- resentation theory of p-adic Lie groups, (4) algebra such as finite simple groups, infinite simple groups, algebraic K-theory and representations of al- gebraic groups over finite fields. By putting together many different types of buildings and applications in seemly unrelated topics, we hope to present an overview of rich structures and applications of buildings and the underlying groups. Keywords. Tits building, Bruhat-Tits building, Bruhat-Tits theory, spherical building, Euclidean, topological building, twin building, arithmetic group, Mostow strong rigidity, Margulis super-rigidity, BN-pair, compacti- fication, quasi-isometry rigidity, Kac-Moody group, CAT(0)-space, CAT(0)- group, Solomon-Tits Theorem, cohomology group, cohomology dimension, generalized Poincar´eduality, simplicial volume, Coxeter complex, incidence geometry, finite geometry, simplicial complex, tree, geometrization conjec- ture of Thurston, curve complex, Teichm¨ullerspace, mapping class group. AMS subject classification numbers. 20E45, 53C24, 11F75. ∗Department of Mathematics, University of Michigan, Ann Arbor, MI 48109. Email address, [email protected] 2 Lizhen Ji Contents 1 Introduction and history of buildings 3 1.1 Summary . 3 1.2 History of buildings and outline of the paper . 5 1.3 Acknowledgments and dedication . 14 2 Spherical Tits buildings 15 2.1 Definition of buildings as chamber complexes and Solomon-Tits the- orem . 15 2.2 Semisimple Lie groups and buildings . 17 2.3 BN-pairs or Tits systems, and buildings . 18 2.4 Other definitions of and approaches to buildings . 19 2.5 Rigidity of Tits buildings . 21 3 Geometric realizations and applications of spherical Tits build- ings 23 3.1 Geodesic compactification of symmetric spaces . 23 3.2 Buildings and compactifications of symmetric spaces . 25 3.3 Topological spherical Tits buildings and Moufang buildings . 27 3.4 Mostow strong rigidity . 30 3.5 Rank rigidity of manifolds of nonpositive curvature . 35 3.6 Rank rigidity for CAT(0)-spaces and CAT(0)-groups . 37 3.7 Classification of isoparametric submanifolds . 41 3.8 Spherical buildings and compactifications of locally symmetric spaces 45 3.9 Geodesic compactification, Gromov compactification and large scale geometry . 49 3.10 Cohomology of arithmetic groups . 50 3.11 Vanishing of simplicial volume of high rank locally symmetric spaces 55 3.12 Generalizations of buildings: curve complexes and applications . 57 4 Euclidean buildings 58 4.1 Definitions and basic properties . 58 4.2 Semisimple p-adic groups and Euclidean buildings . 59 4.3 Compactification of Euclidean buildings by spherical buildings . 63 4.4 Satake compactifications of Bruhat-Tits buildings . 66 5 Applications of Euclidean buildings 67 5.1 p-adic curvature and vanishing of cohomology of lattices . 67 5.2 Super-rigidity and harmonic maps into Euclidean buildings . 69 5.3 Applications to S-arithmetic groups . 74 5.4 Applications to harmonic analysis and representation theories . 78 Buildings and their applications in geometry and topology 3 6 R-trees and R-buildings 80 6.1 Definition of R-trees and basic properties . 81 6.2 Applications of R-trees in topology . 82 6.3 R-Euclidean buildings . 88 6.4 Quasi-isometry rigidity and tangent cones at infinity of symmetric spaces . 89 7 Twin buildings and Kac-Moody groups 90 7.1 Twin buildings . 91 7.2 Kac-Moody algebras and Kac-Moody groups . 92 7.3 Kac-Moody groups as lattices and groups arising from buildings in geometric group theory . 94 8 Other applications of buildings 96 8.1 Applications in algebraic geometry . 96 8.2 Random walks and the Martin boundary . 98 8.3 Finite groups . 99 8.4 Finite geometry . 99 8.5 Algebraic K-groups . 100 8.6 Algebraic combinatorics . 100 8.7 Expanders and Ramanujan graphs . 100 1 Introduction and history of buildings 1.1 Summary Classical simple Lie groups over C are the symmetry groups of quadratic forms or sesquilinear forms of finite dimensional vector spaces over C. The same construc- tion works for vector spaces over finite fields and produces classical finite groups of Lie type, but this method does not extend to exceptional simple Lie groups. Buildings were originally introduced by Tits in 1950s in order to realize excep- tional Lie groups as the symmetry groups of spaces (or geometry) so that one can construct geometrically analogues of exceptional simple Lie groups over arbitrary fields.1 The first buildings were spherical buildings, and they have been generalized in various ways: Euclidean buildings (Bruhat-Tits buildings), topological buildings, R-buildings, in particular R-trees, twin buildings and hyperbolic buildings. Besides the combinatorial properties, they often carry additional structures from geome- try and group theory. They are useful for many different applications in various subjects such as algebraic groups, group theory and in particular finite groups, fi- nite geometry, representation theory over local fields, algebraic geometry, Arakelov 1According to [RT, p. 292], \... it is perhaps worth remarking that one of the initial motiva- tions for the theory of buildings, at a time when Chevalley's fundamental \Tohoku paper" had not yet appeared, was the search for a geometric way of obtaining algebraic analogues of the exceptional Lie groups." 4 Lizhen Ji intersection for arithmetic varieties, algebraic K-theories, combinatorial and geo- metric group theory, global differential geometry, geometric and algebraic topology, cohomology groups of arithmetic groups and S-arithmetic groups, Mostow strong rigidity and Margulis super-rigidity of lattice subgroups of semisimple Lie groups and of nonpositively curved manifolds, classification of isoparametric submani- folds in Rn of high codimension, and existence of hyperbolic structures on three dimensional manifolds in Thurston's geometrization program. Basically, there are two reasons why buildings are useful for questions and problems on semisimple (or more generally reductive) groups and their subgroups, and generalizations such as Kac-Moody groups: 1. For semisimple (or reductive) Lie groups and algebraic groups, buildings pro- vide effective ways to parametrize subgroups such as parabolic subgroups of reductive algebraic groups over any fields and parahoric subgroups of reduc- tive algebraic groups over locally compact, totally disconnected fields, and relations between these subgroups can also be expressed and reflected effec- tively by the geometry of buildings. Hence properties and structures of the groups can be described, understood, and proved via buildings. Rich geom- etry (for example existence of ample collections of subspaces) of buildings is a geometric reflection of rich structure of the associated (parahoric) groups. 2. Buildings also provide spaces on which the algebraic groups and their sub- groups act, and representation spaces of these groups can be constructed and studied. Furthermore, the automorphism groups of locally finite buildings provide natural locally compact topological groups which can be used to understand groups that act on the buildings, for example, the Kac-Moody groups. In this paper, several applications of buildings in differential geometry, geo- metric topology and group cohomology theory will be emphasized. There are four underlying themes in these applications: 1. Spherical buildings often describe the geometry at infinity of symmetric spaces and locally symmetric spaces, and hence are large scale geometric invariants of these spaces. The topology of spherical buildings often de- scribes precisely the topology at infinity of locally symmetric spaces. Both spherical buildings and R-buildings also appear as limiting objects under degeneration or scaling of Riemannian manifolds and more generally metric spaces. 2. Euclidean buildings are analogues of symmetric spaces for semisimple groups defined over local fields and their discrete subgroups, and they play similar roles in understanding properties and structures of algebraic groups and their subgroups, for example, generalization of the standard decompositions such as the Cartan decomposition and Iwasawa decomposition. 3. Euclidean buildings and symmetric spaces of noncompact type provide the most important examples of CAT(0)-spaces, which are simply connected and Buildings and their applications in geometry and topology 5 nonpositively curved geodesic metric spaces. This fact is used crucially in proving the super-rigidity and arithmeticity of co-finite discrete subgroups of the rank one simple Lie group Sp(1; n) and F4(−20), the isometry groups of quaternionic hyperbolic space and the Cayley plane. Furthermore, groups that act properly and isometrically on buildings provide some of the most in- teresting groups in geometric group theory, for example, Kac-Moody groups over finite fields. 4. Spherical buildings of higher rank
Load more

Recommended publications
  • Affine Springer Fibers and Affine Deligne-Lusztig Varieties
    Affine Springer Fibers and Affine Deligne-Lusztig Varieties Ulrich G¨ortz Abstract. We give a survey on the notion of affine Grassmannian, on affine Springer fibers and the purity conjecture of Goresky, Kottwitz, and MacPher- son, and on affine Deligne-Lusztig varieties and results about their dimensions in the hyperspecial and Iwahori cases. Mathematics Subject Classification (2000). 22E67; 20G25; 14G35. Keywords. Affine Grassmannian; affine Springer fibers; affine Deligne-Lusztig varieties. 1. Introduction These notes are based on the lectures I gave at the Workshop on Affine Flag Man- ifolds and Principal Bundles which took place in Berlin in September 2008. There are three chapters, corresponding to the main topics of the course. The first one is the construction of the affine Grassmannian and the affine flag variety, which are the ambient spaces of the varieties considered afterwards. In the following chapter we look at affine Springer fibers. They were first investigated in 1988 by Kazhdan and Lusztig [41], and played a prominent role in the recent work about the “fun- damental lemma”, culminating in the proof of the latter by Ngˆo. See Section 3.8. Finally, we study affine Deligne-Lusztig varieties, a “σ-linear variant” of affine Springer fibers over fields of positive characteristic, σ denoting the Frobenius au- tomorphism. The term “affine Deligne-Lusztig variety” was coined by Rapoport who first considered the variety structure on these sets. The sets themselves appear implicitly already much earlier in the study of twisted orbital integrals. We remark that the term “affine” in both cases is not related to the varieties in question being affine, but rather refers to the fact that these are notions defined in the context of an affine root system.
    [Show full text]
  • Dynamics for Discrete Subgroups of Sl 2(C)
    DYNAMICS FOR DISCRETE SUBGROUPS OF SL2(C) HEE OH Dedicated to Gregory Margulis with affection and admiration Abstract. Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups [30]: \A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: these two theories can be united under the common title Theory of discrete subgroups of SL2(C)". In this article, we will discuss a few recent advances regarding this missing topic from his book, which were influenced by his earlier works. Contents 1. Introduction 1 2. Kleinian groups 2 3. Mixing and classification of N-orbit closures 10 4. Almost all results on orbit closures 13 5. Unipotent blowup and renormalizations 18 6. Interior frames and boundary frames 25 7. Rigid acylindrical groups and circular slices of Λ 27 8. Geometrically finite acylindrical hyperbolic 3-manifolds 32 9. Unipotent flows in higher dimensional hyperbolic manifolds 35 References 44 1. Introduction A discrete subgroup of PSL2(C) is called a Kleinian group. In this article, we discuss dynamics of unipotent flows on the homogeneous space Γn PSL2(C) for a Kleinian group Γ which is not necessarily a lattice of PSL2(C). Unlike the lattice case, the geometry and topology of the associated hyperbolic 3-manifold M = ΓnH3 influence both topological and measure theoretic rigidity properties of unipotent flows. Around 1984-6, Margulis settled the Oppenheim conjecture by proving that every bounded SO(2; 1)-orbit in the space SL3(Z)n SL3(R) is compact ([28], [27]).
    [Show full text]
  • Structure, Classification, and Conformal Symmetry, of Elementary
    Lett Math Phys (2009) 89:171–182 DOI 10.1007/s11005-009-0351-2 Structure, Classification, and Conformal Symmetry, of Elementary Particles over Non-Archimedean Space–Time V. S. VARADARAJAN and JUKKA VIRTANEN Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA. e-mail: [email protected]; [email protected] Received: 3 April 2009 / Revised: 12 September 2009 / Accepted: 12 September 2009 Publishedonline:23September2009–©TheAuthor(s)2009. This article is published with open access at Springerlink.com Abstract. It is known that no length or time measurements are possible in sub-Planckian regions of spacetime. The Volovich hypothesis postulates that the micro-geometry of space- time may therefore be assumed to be non-archimedean. In this letter, the consequences of this hypothesis for the structure, classification, and conformal symmetry of elementary particles, when spacetime is a flat space over a non-archimedean field such as the p-adic numbers, is explored. Both the Poincare´ and Galilean groups are treated. The results are based on a new variant of the Mackey machine for projective unitary representations of semidirect product groups which are locally compact and second countable. Conformal spacetime is constructed over p-adic fields and the impossibility of conformal symmetry of massive and eventually massive particles is proved. Mathematics Subject Classification (2000). 22E50, 22E70, 20C35, 81R05. Keywords. Volovich hypothesis, non-archimedean fields, Poincare´ group, Galilean group, semidirect product, cocycles, affine action, conformal spacetime, conformal symmetry, massive, eventually massive, massless particles. 1. Introduction In the 1970s many physicists, concerned about the divergences in quantum field theories, started exploring the micro-structure of space–time itself as a possible source of these problems.
    [Show full text]
  • Noncommutative Geometry and the Spectral Model of Space-Time
    S´eminaire Poincar´eX (2007) 179 – 202 S´eminaire Poincar´e Noncommutative geometry and the spectral model of space-time Alain Connes IHES´ 35, route de Chartres 91440 Bures-sur-Yvette - France Abstract. This is a report on our joint work with A. Chamseddine and M. Marcolli. This essay gives a short introduction to a potential application in physics of a new type of geometry based on spectral considerations which is convenient when dealing with noncommutative spaces i.e. spaces in which the simplifying rule of commutativity is no longer applied to the coordinates. Starting from the phenomenological Lagrangian of gravity coupled with matter one infers, using the spectral action principle, that space-time admits a fine structure which is a subtle mixture of the usual 4-dimensional continuum with a finite discrete structure F . Under the (unrealistic) hypothesis that this structure remains valid (i.e. one does not have any “hyperfine” modification) until the unification scale, one obtains a number of predictions whose approximate validity is a basic test of the approach. 1 Background Our knowledge of space-time can be summarized by the transition from the flat Minkowski metric ds2 = − dt2 + dx2 + dy2 + dz2 (1) to the Lorentzian metric 2 µ ν ds = gµν dx dx (2) of curved space-time with gravitational potential gµν . The basic principle is the Einstein-Hilbert action principle Z 1 √ 4 SE[ gµν ] = r g d x (3) G M where r is the scalar curvature of the space-time manifold M. This action principle only accounts for the gravitational forces and a full account of the forces observed so far requires the addition of new fields, and of corresponding new terms SSM in the action, which constitute the Standard Model so that the total action is of the form, S = SE + SSM .
    [Show full text]
  • ADELIC VERSION of MARGULIS ARITHMETICITY THEOREM Hee Oh 1. Introduction Let R Denote the Set of All Prime Numbers Including
    ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM Hee Oh Abstract. In this paper, we generalize Margulis’s S-arithmeticity theorem to the case when S can be taken as an infinite set of primes. Let R be the set of all primes including infinite one ∞ and set Q∞ = R. Let S be any subset of R. For each p ∈ S, let Gp be a connected semisimple adjoint Qp-group without any Qp-anisotropic factors and Dp ⊂ Gp(Qp) be a compact open subgroup for almost all finite prime p ∈ S. Let (GS , Dp) denote the restricted topological product of Gp(Qp)’s, p ∈ S with respect to Dp’s. Note that if S is finite, (GS , Dp) = Qp∈S Gp(Qp). We show that if Pp∈S rank Qp (Gp) ≥ 2, any irreducible lattice in (GS , Dp) is a rational lattice. We also present a criterion on the collections Gp and Dp for (GS , Dp) to admit an irreducible lattice. In addition, we describe discrete subgroups of (GA, Dp) generated by lattices in a pair of opposite horospherical subgroups. 1. Introduction Let R denote the set of all prime numbers including the infinite prime ∞ and Rf the set of finite prime numbers, i.e., Rf = R−{∞}. We set Q∞ = R. For each p ∈ R, let Gp be a non-trivial connected semisimple algebraic Qp-group and for each p ∈ Rf , let Dp be a compact open subgroup of Gp(Qp). The adele group of Gp, p ∈ R with respect to Dp, p ∈ Rf is defined to be the restricted topological product of the groups Gp(Qp) with respect to the distinguished subgroups Dp.
    [Show full text]
  • Connections Between Graph Theory, Additive Combinatorics, and Finite
    UNIVERSITY OF CALIFORNIA, SAN DIEGO Connections between graph theory, additive combinatorics, and finite incidence geometry A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Michael Tait Committee in charge: Professor Jacques Verstra¨ete,Chair Professor Fan Chung Graham Professor Ronald Graham Professor Shachar Lovett Professor Brendon Rhoades 2016 Copyright Michael Tait, 2016 All rights reserved. The dissertation of Michael Tait is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2016 iii DEDICATION To Lexi. iv TABLE OF CONTENTS Signature Page . iii Dedication . iv Table of Contents . .v List of Figures . vii Acknowledgements . viii Vita........................................x Abstract of the Dissertation . xi 1 Introduction . .1 1.1 Polarity graphs and the Tur´annumber for C4 ......2 1.2 Sidon sets and sum-product estimates . .3 1.3 Subplanes of projective planes . .4 1.4 Frequently used notation . .5 2 Quadrilateral-free graphs . .7 2.1 Introduction . .7 2.2 Preliminaries . .9 2.3 Proof of Theorem 2.1.1 and Corollary 2.1.2 . 11 2.4 Concluding remarks . 14 3 Coloring ERq ........................... 16 3.1 Introduction . 16 3.2 Proof of Theorem 3.1.7 . 21 3.3 Proof of Theorems 3.1.2 and 3.1.3 . 23 3.3.1 q a square . 24 3.3.2 q not a square . 26 3.4 Proof of Theorem 3.1.8 . 34 3.5 Concluding remarks on coloring ERq ........... 36 4 Chromatic and Independence Numbers of General Polarity Graphs .
    [Show full text]
  • A Quasideterminantal Approach to Quantized Flag Varieties
    A QUASIDETERMINANTAL APPROACH TO QUANTIZED FLAG VARIETIES BY AARON LAUVE A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Mathematics Written under the direction of Vladimir Retakh & Robert L. Wilson and approved by New Brunswick, New Jersey May, 2005 ABSTRACT OF THE DISSERTATION A Quasideterminantal Approach to Quantized Flag Varieties by Aaron Lauve Dissertation Director: Vladimir Retakh & Robert L. Wilson We provide an efficient, uniform means to attach flag varieties, and coordinate rings of flag varieties, to numerous noncommutative settings. Our approach is to use the quasideterminant to define a generic noncommutative flag, then specialize this flag to any specific noncommutative setting wherein an amenable determinant exists. ii Acknowledgements For finding interesting problems and worrying about my future, I extend a warm thank you to my advisor, Vladimir Retakh. For a willingness to work through even the most boring of details if it would make me feel better, I extend a warm thank you to my advisor, Robert L. Wilson. For helpful mathematical discussions during my time at Rutgers, I would like to acknowledge Earl Taft, Jacob Towber, Kia Dalili, Sasa Radomirovic, Michael Richter, and the David Nacin Memorial Lecture Series—Nacin, Weingart, Schutzer. A most heartfelt thank you is extended to 326 Wayne ST, Maria, Kia, Saˇsa,Laura, and Ray. Without your steadying influence and constant comraderie, my time at Rut- gers may have been shorter, but certainly would have been darker. Thank you. Before there was Maria and 326 Wayne ST, there were others who supported me.
    [Show full text]
  • Finite Projective Geometries 243
    FINITE PROJECTÎVEGEOMETRIES* BY OSWALD VEBLEN and W. H. BUSSEY By means of such a generalized conception of geometry as is inevitably suggested by the recent and wide-spread researches in the foundations of that science, there is given in § 1 a definition of a class of tactical configurations which includes many well known configurations as well as many new ones. In § 2 there is developed a method for the construction of these configurations which is proved to furnish all configurations that satisfy the definition. In §§ 4-8 the configurations are shown to have a geometrical theory identical in most of its general theorems with ordinary projective geometry and thus to afford a treatment of finite linear group theory analogous to the ordinary theory of collineations. In § 9 reference is made to other definitions of some of the configurations included in the class defined in § 1. § 1. Synthetic definition. By a finite projective geometry is meant a set of elements which, for sugges- tiveness, are called points, subject to the following five conditions : I. The set contains a finite number ( > 2 ) of points. It contains subsets called lines, each of which contains at least three points. II. If A and B are distinct points, there is one and only one line that contains A and B. HI. If A, B, C are non-collinear points and if a line I contains a point D of the line AB and a point E of the line BC, but does not contain A, B, or C, then the line I contains a point F of the line CA (Fig.
    [Show full text]
  • Twisted Loop Groups and Their Affine Flag Varieties
    TWISTED LOOP GROUPS AND THEIR AFFINE FLAG VARIETIES G. PAPPAS* AND M. RAPOPORT Introduction Loop groups are familiar objects in several branches of mathematics. Let us mention here three variants. The first variant is differential-geometric in nature. One starts with a Lie group G (e.g., a compact Lie group or its complexification). The associated loop group is then the group of (C0-, or C1-, or C∞-) maps of S1 into G, cf. [P-S] and the literature cited there. A twisted version arises from an automorphism α of G. The associated twisted loop group is the group of maps γ : R → G such that γ(θ + 2π) = α(γ(θ)) . The second variant is algebraic and arises in the context of Kac-Moody algebras. Here one constructs an infinite-dimensional algebraic group variety with Lie algebra equal or closely related to a given Kac-Moody algebra. (This statement is an oversimplification and the situation is in fact more complicated: there exist various constructions at a formal, a minimal, and a maximal level which produce infinite-dimensional groups with Lie algebras closely related to the given Kac-Moody Lie algebra, see [Ma2], also [T2], [T3] and the literature cited there). The third variant is algebraic-geometric in nature and is our main concern in this paper. Let us recall the basic definitions in the untwisted case. Let k be a field and let G0 be an algebraic group over Spec (k). We consider the functor LG0 on the category of k-algebras, R 7→ LG0(R) = G0(R((t))).
    [Show full text]
  • On Some Recent Developments in the Theory of Buildings
    On some recent developments in the theory of buildings Bertrand REMY∗ Abstract. Buildings are cell complexes with so remarkable symmetry properties that many groups from important families act on them. We present some examples of results in Lie theory and geometric group theory obtained thanks to these highly transitive actions. The chosen examples are related to classical and less classical (often non-linear) group-theoretic situations. Mathematics Subject Classification (2010). 51E24, 20E42, 20E32, 20F65, 22E65, 14G22, 20F20. Keywords. Algebraic, discrete, profinite group, rigidity, linearity, simplicity, building, Bruhat-Tits' theory, Kac-Moody theory. Introduction Buildings are cell complexes with distinguished subcomplexes, called apartments, requested to satisfy strong incidence properties. The notion was invented by J. Tits about 50 years ago and quickly became useful in many group-theoretic situations [75]. By their very definition, buildings are expected to have many symmetries, and this is indeed the case quite often. Buildings are relevant to Lie theory since the geometry of apartments is described by means of Coxeter groups: apartments are so to speak generalized tilings, where a usual (spherical, Euclidean or hyper- bolic) reflection group may be replaced by a more general Coxeter group. One consequence of the existence of sufficiently large automorphism groups is the fact that many buildings admit group actions with very strong transitivity properties, leading to a better understanding of the groups under consideration. The beginning of the development of the theory is closely related to the theory of algebraic groups, more precisely to Borel-Tits' theory of isotropic reductive groups over arbitrary fields and to Bruhat-Tits' theory of reductive groups over non-archimedean valued fields.
    [Show full text]
  • FINITE GROUP ACTIONS on REDUCTIVE GROUPS and BUILDINGS and TAMELY-RAMIFIED DESCENT in BRUHAT-TITS THEORY by Gopal Prasad Dedicat
    FINITE GROUP ACTIONS ON REDUCTIVE GROUPS AND BUILDINGS AND TAMELY-RAMIFIED DESCENT IN BRUHAT-TITS THEORY By Gopal Prasad Dedicated to Guy Rousseau Abstract. Let K be a discretely valued field with Henselian valuation ring and separably closed (but not necessarily perfect) residue field of characteristic p, H a connected reductive K-group, and Θ a finite group of automorphisms of H. We assume that p does not divide the order of Θ and Bruhat-Tits theory is available for H over K with B(H=K) the Bruhat-Tits building of H(K). We will show that then Bruhat-Tits theory is also available for G := (HΘ)◦ and B(H=K)Θ is the Bruhat-Tits building of G(K). (In case the residue field of K is perfect, this result was proved in [PY1] by a different method.) As a consequence of this result, we obtain that if Bruhat-Tits theory is available for a connected reductive K-group G over a finite tamely-ramified extension L of K, then it is also available for G over K and B(G=K) = B(G=L)Gal(L=K). Using this, we prove that if G is quasi-split over L, then it is already quasi-split over K. Introduction. This paper is a sequel to our recent paper [P2]. We will assume fa- miliarity with that paper; we will freely use results, notions and notations introduced in it. Let O be a discretely valued Henselian local ring with valuation !. Let m be the maximal ideal of O and K the field of fractions of O.
    [Show full text]
  • Matrices Lie: an Introduction to Matrix Lie Groups and Matrix Lie Algebras
    Matrices Lie: An introduction to matrix Lie groups and matrix Lie algebras By Max Lloyd A Journal submitted in partial fulfillment of the requirements for graduation in Mathematics. Abstract: This paper is an introduction to Lie theory and matrix Lie groups. In working with familiar transformations on real, complex and quaternion vector spaces this paper will define many well studied matrix Lie groups and their associated Lie algebras. In doing so it will introduce the types of vectors being transformed, types of transformations, what groups of these transformations look like, tangent spaces of specific groups and the structure of their Lie algebras. Whitman College 2015 1 Contents 1 Acknowledgments 3 2 Introduction 3 3 Types of Numbers and Their Representations 3 3.1 Real (R)................................4 3.2 Complex (C).............................4 3.3 Quaternion (H)............................5 4 Transformations and General Geometric Groups 8 4.1 Linear Transformations . .8 4.2 Geometric Matrix Groups . .9 4.3 Defining SO(2)............................9 5 Conditions for Matrix Elements of General Geometric Groups 11 5.1 SO(n) and O(n)........................... 11 5.2 U(n) and SU(n)........................... 14 5.3 Sp(n)................................. 16 6 Tangent Spaces and Lie Algebras 18 6.1 Introductions . 18 6.1.1 Tangent Space of SO(2) . 18 6.1.2 Formal Definition of the Tangent Space . 18 6.1.3 Tangent space of Sp(1) and introduction to Lie Algebras . 19 6.2 Tangent Vectors of O(n), U(n) and Sp(n)............. 21 6.3 Tangent Space and Lie algebra of SO(n).............. 22 6.4 Tangent Space and Lie algebras of U(n), SU(n) and Sp(n)..
    [Show full text]