Armand Borel

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Author Description

Armand Borel (1923–2003) was a Swiss-American mathematician renowned for his foundational work in the theory of Lie groups and their algebraic aspects, as well as for his contributions to algebraic topology. Here are some key points about his life and work:

• Background and Education: Borel was born in La Chaux-de-Fonds, Switzerland. He studied in Geneva and then did advanced research at institutions such as the École Normale Supérieure in Paris.

• Lie Groups and Algebraic Groups: Borel made seminal contributions to the theory of linear algebraic groups, including the concepts of Borel subgroups and Borel measures, which bear his name. His research greatly influenced how mathematicians understand symmetries in geometry and other fields.

• Bourbaki Group: He was a member of the French mathematical collective Nicolas Bourbaki, known for its rigorous and unified presentation of mathematics.

• Institute for Advanced Study: Borel spent much of his career at the Institute for Advanced Study in Princeton, New Jersey, where he served on the faculty and influenced generations of mathematicians.

• Publications and Legacy: He authored numerous influential textbooks and research papers, shaping modern approaches to group theory, topology, and geometry. His work remains central in areas spanning number theory, representation theory, and more.

Overall, Armand Borel is remembered as one of the leading figures of twentieth-century mathematics for his substantial impact on Lie theory and for mentoring many prominent mathematicians.

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Linear algebraic groups

1966-01-01

Armand Borel

Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions. Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.

Linear Algebraic Groups

1991-01-01

Armand Borel

Sur La Cohomologie des Espaces Fibres Principaux et des Espaces Homogenes de Groupes de Lie Compacts

1953-01-01

Armand Borel

Arithmetic Subgroups of Algebraic Groups

1962-05-01

Armand Borel , Harish-chandra Harish-Chandra

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Seminar on Algebraic Groups and Related Finite Groups

1970-01-01

Armand Borel , Rayna J. Carter , C. W. Curtis +3 more

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Corners and arithmetic groups

1973-12-01

Armand Borel , J-P. Serre

Compactification of Arithmetic Quotients of Bounded Symmetric Domains

1966-11-01

Walter L. Baily , Armand Borel

Stable Real Cohomology of Arithmetic Groups II

1981-01-01

Armand Borel

Given a discrete subgroup Γ of a connected real semisimple Lie group G with finite center there is a natural homomorphism (1) % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 … Given a discrete subgroup Γ of a connected real semisimple Lie group G with finite center there is a natural homomorphism (1) % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAamaaDa % aaleaacqqHtoWraeaacaWGXbaaaOGaaiOoaiaadMeadaqhaaWcbaGa % am4raaqaaiaadghaaaGccqGHsgIRcaWGibWaaWbaaSqabeaacaWGXb % aaaOWaaeWaaeaacqqHtoWrcaGG7aGaam4yaaGaayjkaiaawMcaaiaa % ywW7daqadaqaaiaadghacqGH9aqpcaaIWaGaaiilaiaaigdacaGGSa % GaeSOjGSeacaGLOaGaayzkaaGaaiilaaaa!4F34! $$j_\Gamma ^q:I_G^q \to {H^q}\left( {\Gamma ;c} \right)\quad \left( {q = 0,1, \ldots } \right),$$ where I G q denotes the space of G-invariant harmonic q-forms on the symmetric space quotient X=G/K of G by a maximal compact subgroup K. If Γ is cocompact, this homomorphism is injective in all dimensions and the main objective of Matsushima in [19] is to give a range m(G), independent of Γ, in which j Γ q is also surjective. The main argument there is to show that if a certain quadratic form depending on q is positive non-degenerate, then any Γ-invariant harmonic q-form is automatically G-invariant. In [3], we proved similarly the existence of a range in which j Γ q is bijective when Γ is arithmetic, but not necessarily cocompact. There are three main steps to the proof: (i) The cohomology of Γ can be computed by using differential forms which satisfy a certain growth condition, "logarithmic growth," at infinity; (ii) up to some range c(G), these forms are all square integrable; and (iii) use the fact, pointed out in [16] , that for q ≦ m(G), Matsushima's arguments remain valid in the non-compact case for square integrable forms.

Compact Clifford-Klein forms of symmetric spaces

1963-01-01

Armand Borel

Algebraic D-modules

1987-01-01

Armand Borel

Les sous-groupes fermés de rang maximum des groupes de Lie clos

1949-12-01

Armand Borel , Jean Siebenthal

Introduction aux groupes arithmétiques

1969-01-01

Armand Borel

Stable real cohomology of arithmetic groups

1974-01-01

Armand Borel

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Homomorphismes "Abstraits" de Groupes Algebriques Simples

1973-05-01

Armand Borel , Jacques Tits

�l�ments unipotents et sous-groupes paraboliques de groupes r�ductifs. I

1971-06-01

Armand Borel , Jacques Tits

Le théorème de Riemann-Roch

1958-01-01

Armand Borel , Jean-Pierre Serre

d'après des résultats inédits de A. GROTHENDIECK). INTRODUCTION.Ce qui suit constitue les notes d'un séminaire tenu à Princeton en automne 19^7 sur les travaux de GROTHENDIECK; les résultats nouveaux qui … d'après des résultats inédits de A. GROTHENDIECK). INTRODUCTION.Ce qui suit constitue les notes d'un séminaire tenu à Princeton en automne 19^7 sur les travaux de GROTHENDIECK; les résultats nouveaux qui y figurent sont dus à ce dernier; notre contribution est uniquement de nature rédactionnelle.Le « théorème de Riemann-Roch » dont il s'agit est valable pour des variétés algébriques (non singulières) sur un corps de caratéristique quelconque; dans le cas classique, où le corps de base est G, ce théorème contient comme cas particulier celui démontré il y a quelques années parLa démonstration proprement dite du théorème de Riemann-Roch occupe les paragraphes 7 à 16, le dernier paragraphe étant consacré à une application.Les paragraphes 1 à 6 contiennent des préliminaires sur les faisceaux algébriques cohérents [cf.FAC ( 1 )].La terminologie suivie est celle de FAC, à une différence près : pour nous conformer à une coutume qui se répand de plus en plus, nous avons appelé « morphismes » les « applications régulières » de FAC.

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Kählerian Coset Spaces of Semisimple Lie Groups

1954-12-01

Armand Borel

facts for Fourier expansions on compact group and homogeneous spaces.For summability at points see facts for Fourier expansions on compact group and homogeneous spaces.For summability at points see

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Admissible representations of a semi-simple group over a local field with vectors fixed under an iwahori subgroup

1976-12-01

Armand Borel

La classe d'homologie fondamentale d'un espace analytique

1961-01-01

Armand Borel , André Haefliger

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Théorèmes de finitude en cohomologie galoisienne

1964-12-01

Armand Borel , Jean-Pierre Serre

Density Properties for Certain Subgroups of Semi-Simple Groups Without Compact Components

1960-07-01

Armand Borel

Some finiteness properties of adele groups over number fields

1963-12-01

Armand Borel

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Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem

1972-01-01

Armand Borel

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Sur L'Homologie et la Cohomologie des Groupes de Lie Compacts Connexes

1954-04-01

Armand Borel

Groupes de Lie et Puissances Reduites de Steenrod

1953-07-01

Armand Borel , Jean-Pierre Serre

Homology theory for locally compact spaces.

1960-01-01

Armand Borel , John C. Moore

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Properties and linear representations of Chevalley groups

1970-01-01

Armand Borel

Topology of Lie groups and characteristic classes

1955-01-01

Armand Borel

Introduction.The notion of continuous group, later called Lie group, introduced by S. Lie in the nineteenth century, has classically a local character.Although global Lie groups were also sometimes considered, it … Introduction.The notion of continuous group, later called Lie group, introduced by S. Lie in the nineteenth century, has classically a local character.Although global Lie groups were also sometimes considered, it is only after 1920 that this concept was clearly formulated.We recall that a Lie group in the large is first a manifold, i.e., a topological Hausdorff space admitting a covering by open sets, each of which is homeomorphic to euclidian w-space; second it is a group; third it is a topological group, i.e., the product x-y of x and y and the inverse x" 1 are continuous functions of their arguments; and finally it is required that there exist coordinates in a neighbourhood V of the identity element e such that if x, y, and x-y are in V, the coordinates of x-y are analytic functions in the coordinates of x and y.Gleason, Montgomery, and Zippin recently proved that the last condition follows from the others, thus solving Hilbert's fifth problem, with which we shall not be concerned here.As soon as the concept was defined with precision, there arose the problem of studying topological properties of such group-manifolds.Indeed, in the first paper which systematically considers global Lie groups, H. Weyl's famous paper on linear representations [8l], a key result which states that the fundamental group of a compact semisimple Lie group is finite is topological in nature.The question was next considered by E. Cartan in several papers, and later on by many mathematicians; as a matter of fact, it was often generalized in order to include also the study of "homogeneous spaces," i.e., manifolds which admit a transitive Lie group of homeomorphisms.A very complete survey of the work done in this field up to 1951 has been published in this Bulletin by H. Samelson [67].Although of course some overlap is unavoidable, the present report is meant as a sequel and will therefore concentrate mainly on developments which occurred during these very last years.It will be devoted for the greater An address delivered before the New York meeting of the Society on February 27, 1954 by invitation of the Committee to Select Hour Speakers for Eastern Sectional Meetings; received by the editors May 7, 1955.1 This report also surveys material expounded at the Summer Mathematical Institute on Lie groups and Lie algebras, Colby College, 1953.The author expresses his hearty thanks to Dr. W. G. Lister, who prepared

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Algebraic groups and Discontinuous Subgroups

1966-01-01

Armand Borel , G. D. Mostow

Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes

1961-01-01

Armand Borel

Ce travail apporte quelques renseignements sur la torsion du groupe de cohomologie entiere d'un groupe de Lie compact connexe G, que nous appellerons, suivant Γusage, la torsion de G, sur … Ce travail apporte quelques renseignements sur la torsion du groupe de cohomologie entiere d'un groupe de Lie compact connexe G, que nous appellerons, suivant Γusage, la torsion de G, sur les sous-groupes commutatifs de G, et met ces deux questions en relation.Nous nous interesserons en particulier aux rapports qui existent entre la ^-torsion (p nombre premier) et les sous-groupes commutatifs de type (p, ,p), que nous appellerons ici les [^>]-sous-groupes.Ce travail a ete resume dans une Note au Bull.Amer.Math.Soc.66 (I960),.pp.285-288.En nous appuyant sur quelques remarques concernant les ί/-espaces dont la cohomologie entiere est de type fini, faites dans le §1, et sur le Theoreme V de [14], on verra que le groupe simplement connexe exceptionnel E έ n'a pas de ^-torsion lorsque p = 5, i = 6, 7 et p = 7, i = 7, 8, et Ton determinera aussi if* (E 6 ; Z 3 ), H* (E 8 ; Z 5 ), (Theor.2.2,2.3).Compte tenu de resultats connus [5,8,14], on pourra alors indiquer les nombres premiers intervenant dans les coefficients de torsion de tous les groupes simples et simplement connexes (2.5) et Γon verifiera (2.6) le theoreme suivant, conjecture dans [7]: THEOREME A. Supposons G, simple et simplement connexe, et soit p un nombre premier ne divisant pas les coefficients de la plus grande racine de G, exprimee comme somme de racines simples.Alors G na pas de p-torsion.Dans un groupe de Lie compact connexe G il existe des [/>]-sous-groupes evidents, les elements d'ordre p d'un tore maximal, mais il peut y en avoir d'autres, (pour p = 2, les matrices diagonales de SO (n), (n > 3) par exemple).Cependant, d'apres [9, XII,5,3, 5.4], si G n'a pas de p-torsion, tout [/>]-sousgroupe // fait partie d'un tore, ce qui precise un resultat de [11] affirmant que,.sous Γhypothese faite, le rang de H est au plus egal a la dimension des tores maximaux de G. Cette derniere condition est evidemment necessaire pour que H fasse partie d'un tore de G, mais elle n'est pas suffisante en general.En effet, on demontrera: (i) Pour que le groupe fundamental TΓ^G) de G soit sans p-torsion, il faut et il suffit que tout [/>]-sous-groupe de rang deux soit contenu dans un

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Rationality properties of linear algebraic groups, II

1968-01-01

Armand Borel , T. A. Springer

This paper is a development of [4], and gives a more detailed treatment of the topic named in the title.It includes in particular the birational equivalence with affine space, over … This paper is a development of [4], and gives a more detailed treatment of the topic named in the title.It includes in particular the birational equivalence with affine space, over the groundfield, of the variety of Cartan subgroups of a k-group G, the splitting of G over a separable extension of k if G is reductive, some results on unipotent groups operated upon by tori, and on the existence of subgroups of G whose Lie algebra contains a given nilpotent element of the Lie algebra g of G.Discussing as it does a number of known results (due mostly to Rosenlicht and Grothendieck), this paper is to be viewed as partly expository.In fact, besides proving some new results, our main goal is to provide a rather comprehensive, albeit not exhaustive, account of our topic, from the point of view sketched in [4],Our basic tools are some rationality properties of transversal intersections and of separable mappings, the Jordan decomposition in g, and purely inseparable isogenies of height one.They are reviewed or discussed in section 1.13, §3 and §5 respectively.Thus Lie algebras of algebraic groups play an important role in this paper and, for the sake of completeness, we have collected in §1 a number of definitions and facts pertaining to them.§2 reproves a result of Grothendieck ([12], Exp.XIV) stating that g is the union of the subalgebras of its Borel subgroups.Its main use for us is to reduce to Lie algebras of solvable groups the existence proof of the Jordan decomposition, §4 discusses subalgebras § of g consisting of semi-simple elements, to be called "toral subalgebras" of g.They are tangent to maximal tori, and have several properties similar to that of tori in G, in particular: the centralizer Z( 8)={g*G 9 Adg(X) = X(X € 8)} of jg in G is defined over k if 3 is, (see 4.3 for Z(3)°, 6.14 for Z(β)\ its Lie algebra is 3(8) = {X z g, [8, X] = 0}.If § is spanned by one element X, the conjugacy class of X is isomorphic to G/Z(β).This paragraph also gives some conditions under which a subalgebra of g is algebraic, and reproves some results of Chevalley [8] in characteristic zero.§6 introduces regular elements, Cartan subalgebras in g, and the subgroups of type (C) of ([12], Exp.XΠI) in G.By definition here, a Cartan subalgebra

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Some remarks about Lie groups transitive on spheres and tori

1949-01-01

Armand Borel

The w-dimensional torus as a homogeneous space.In [l], D. Montgomery and H. Samelson proved that a Lie group which acts transitively and effectively on the n-dimensional torus is itself the … The w-dimensional torus as a homogeneous space.In [l], D. Montgomery and H. Samelson proved that a Lie group which acts transitively and effectively on the n-dimensional torus is itself the ^-dimensional toral group T n .Actually, as they remark at the end of

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Intersection Cohomology

1984-01-01

Armand Borel

This volume contains the Notes of a seminar on Intersection Ho- logy which met weekly during the Spring 1983 at the University of Bern, Switzerland. Its main purpose was to … This volume contains the Notes of a seminar on Intersection Ho- logy which met weekly during the Spring 1983 at the University of Bern, Switzerland. Its main purpose was to give an introduction to the

Density and maximality of arithmetic subgroups.

1966-11-01

Armand Borel

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�ber kompakte homogene K�hlersche Mannigfaltigkeiten

1962-10-01

Armand Borel , Reinhold Remmert

Compactifications of Symmetric and Locally Symmetric Spaces

2006-01-01

Armand Borel , Lizhen Ji

Noncompact symmetric and locally symmetric spaces naturally appear in many mathematical theories, including analysis (representation theory, nonabelian harmonic analysis), number theory (automorphic f Noncompact symmetric and locally symmetric spaces naturally appear in many mathematical theories, including analysis (representation theory, nonabelian harmonic analysis), number theory (automorphic f

Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups

1989-12-01

Armand Borel , Gopal Prasad

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La cohomologie mod 2 de certains espaces homogènes

1953-12-01

Armand Borel

Almost commuting elements in compact Lie groups

2002-01-01

Armand Borel , Robert Friedman , John W. Morgan

1.5.1 for C cyclic . . . . . . . . . . . . . 1.5.1 for C cyclic . . . . . . . . . . . . .

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Topics in the Homology Theory of Fibre Bundles

1967-01-01

Armand Borel

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Compactifications of symmetric spaces

2007-01-01

Armand Borel , Liang Ji

Compactifications of symmetric spaces have been constructed by different methods for various applications. One application is to provide the so-called rational boundary components which can be used to compactify locally … Compactifications of symmetric spaces have been constructed by different methods for various applications. One application is to provide the so-called rational boundary components which can be used to compactify locally symmetric spaces. In this paper, we construct many compactifications of symmetric spaces using a uniform method, which is motivated by the Borel-Serre compactification of locally symmetric spaces. Besides unifying compactifications of both symmetric and locally symmetric spaces, this uniform construction allows one to compare and relate easily different compactifications, to extend the group action continuously to boundaries of compactifications, and to clarify the structure of the boundaries.

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Cohomologie d'immeubles et de groupes s-arithmétiques

1976-01-01

Armand Borel , Jean-Pierre Serre

On Semi-Simple Automorphisms of Lie Algebras

1955-05-01

Armand Borel , G. D. Mostow

Cohomologie de $SL_n$ et valeurs de fonctions zêta aux points entiers

1977-01-01

Armand Borel

Sur certains sous-groupes des groupes de Lie compacts

1953-12-01

Armand Borel , Jean-Pierre Serre

The Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics (1869-1926)

2001-11-01

Armand Borel , Thomas Hawkins

I: Sophus Lie.- 1. The Geometrical Origins of Lie's Theory.- 1.1. Tetrahedral Line Complexes.- 1.2. W-Curves and W-Surfaces.- 1.3. Lie's Idee Fixe.- 1.4. The Sphere Mapping.- 1.5. The Erlanger Programm.- … I: Sophus Lie.- 1. The Geometrical Origins of Lie's Theory.- 1.1. Tetrahedral Line Complexes.- 1.2. W-Curves and W-Surfaces.- 1.3. Lie's Idee Fixe.- 1.4. The Sphere Mapping.- 1.5. The Erlanger Programm.- 2. Jacobi and the Analytical Origins of Lie's Theory.- 2.1. Jacobi's Two Methods.- 2.2. The Calculus of Infinitesimal Transformations.- 2.3. Function Groups.- 2.4. The Invariant Theory of Contact Transformations.- 2.5. The Birth of Lie's Theory of Groups.- 3. Lie's Theory of Transformation Groups 1874-1893..- 3.1. The Group Classification Problem.- 3.2. An Overview of Lie's Theory.- 3.3. The Adjoint Group.- 3.4. Complete Systems and Lie's Idee Fixe.- 3.5. The Symplectic Groups.- II: Wilhelm Killing.- 4. The Background to Killing's Work on Lie Algebras.- 4.1. Non-Euclidean Geometry and Weierstrassian Mathematics.- 4.2. Student Years in Berlin: 1867-1872.- 4.3. Non-Euclidean Geometry and General Space Forms.- 4.4. From Space Forms to Lie Algebras.- 4.5. Riemann and Helmholz.- 4.6. Killing and Klein on the Scope of Geometry.- >Chapter 5. Killing and the Structure of Lie Algebrass.- 5.1. Spaces Forms and Characteristic Equations.- 5.2. Encounter with Lie's Theory.- 5.3. Correspondence with Engel.- 5.4. Killing's Theory of Structure.- 5.5. Groups of Rank Zero.- 5.6. The Lobachevsky Prize.- III: Elie Cartan.- 6. The Doctoral Thesis of Elie Cartan.- 6.1. Lie and the Mathematicians of Paris.- 6.2. Cartan's Theory of Semisimple Algebras.- 6.3. Killing's Secondary Roots.- 6.4. Cartan's Application of Secondary Roots.- 7. Lie's School & Linear Representations.- 7.1. Representations in Lie's Research Program.- 7.2. Eduard Study.- 7.3. Gino Fano.- 7.4. Cayley's Counting Problem.- 7.5. Kowalewski's Theory of Weights.- 8. Cartan's Trilogy: 1913-14.- 8.1. Research Priorities 1893-1909.- 8.2. Another Application of Secondary Roots.- 8.3. Continuous Groups and Geometry.- 8.4. The Memoir of 1913.- 8.5. The Memoirs of 1914.- IV: Hermann Weyl.- 9. The Gottingen School of Hilbert.- 9.1. Hilbert and the Theory of Invariants.- 9.2. Hilbert at Gottingen.- 9.3. The Mathematization of Physics at Gottingen ..- 9.4. Weyl's Gottingen Years: Integral Equations.- 9.5. Weyl's Gottingen Years: Riemann Surfaces.- 9.6. Hilbert's Brand of Mathematical Thinking.- 10. The Berlin Algebraists: Frobenius & Schur.- 10.1. Frobenius' Theory of Group Characters & Representations.- 10.2. Hurwitz and the Theory of Invariants.- 10.3. Schur's Doctoral Dissertation.- 10.4. Schur's Career 1901-1923.- 10.5. Cayley's Counting Problem Revisited.- 11. From Relativity to Representations.- 11.1. Einstein's General Theory of Relativity.- 11.2. The Space Problem Reconsidered.- 11.3. Tensor Algebra & Tensor Symmetries.- 11.4. Weyl's Response to Study.- 11.5. The Group-Theoretic Foundation of Tensor Calculus.- 12. Weyl's Great Papers of 1925 and 1926.- 12.1. The Complete Reducibility Theorem.- 12.2. Schur and the Origins of Weyl's 1925 Paper.- 12.3. Weyl's Extension of the Killing-Cartan Theory.- 12.4. Weyl's Finite Basis Theorem.- 12.5. Weyl's Theory of Characters.- 12.6. Cartan's Response.- 12.7. The Peter-Weyl Paper.- Afterword. Suggested Further Reading.- References. Published & Unpublished Sources.

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Arithmetic Subgroups of Algebraic Groups

1984-01-01

Armand Borel , Harish-chandra Harish-Chandra

L2-cohomology of locally symmetric manifolds of finite volume

1983-09-01

Armand Borel , W. Casselman

Laplacian and the Discrete Spectrum of an Arithmetic Group

1983-04-01

Armand Borel , Howard Garland

On the Curvature Tensor of the Hermitian Symmetric Manifolds

1960-05-01

Armand Borel

Introduction to Arithmetic Groups

2019-11-07

Armand Borel

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Algebraic groups

2019-11-07

Armand Borel

On “abstract” homomorphisms of simple algebraic groups

2018-12-31

Armand Borel , Jacques Tits

[80] Originally published in Algebraic geometry: papers presented at the Bombay Colloquium 1968, Tata Inst. Fund. Res. Stud. Math. 4, Oxford University Press, Oxford (1969), 75–82. Reused with permission. [80] Originally published in Algebraic geometry: papers presented at the Bombay Colloquium 1968, Tata Inst. Fund. Res. Stud. Math. 4, Oxford University Press, Oxford (1969), 75–82. Reused with permission.

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On cohomology of principal fiber bundles and homogeneous spaces of compact Lie groups

2012-09-01

Armand Borel

Cohomology mod 2 of some homogeneous spaces

2012-09-01

Armand Borel

Automorphic forms on SL[2](R)

2008-01-01

Armand Borel

Part I. Basic Material On SL2(R), Discrete Subgroups and the Upper-Half Plane: 1. Prerequisites and notation 2. Review of SL2(R), differential operators, convolution 3. Action of G on X, discrete … Part I. Basic Material On SL2(R), Discrete Subgroups and the Upper-Half Plane: 1. Prerequisites and notation 2. Review of SL2(R), differential operators, convolution 3. Action of G on X, discrete subgroups of G, fundamental domains 4. The unit disc model Part II. Automorphic Forms and Cusp Forms: 5. Growth conditions, automorphic forms 6. Poincare series 7. Constant term:the fundamental estimate 8. Finite dimensionality of the space of automorphic forms of a given type 9. Convolution operators on cuspidal functions Part III. Eisenstein Series: 10. Definition and convergence of Eisenstein series 11. Analytic continuation of the Eisenstein series 12. Eisenstein series and automorphic forms orthogonal to cusp forms Part IV. Spectral Decomposition and Representations: 13.Spectral decomposition of L2(G\G)m with respect to C 14. Generalities on representations of G 15. Representations of SL2(R) 16. Spectral decomposition of L2(G\SL2(R)):the discrete spectrum 17. Spectral decomposition of L2(G\SL2(R)): the continuous spectrum 18. Concluding remarks.

Automorphic Forms on Reductive Groups

2007-05-02

Armand Borel

Compactifications of symmetric spaces

2007-01-01

Armand Borel , Liang Ji

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Formes automorphes et séries de dirichlet

2006-11-24

Armand Borel

Cohomologie de certains groupes discretes et lapiacien p-adique [d'après H. Garland]

2006-11-23

Armand Borel

Sous-groupes discrets de groupes semi-simples

2006-11-23

Armand Borel

Compactifications of Locally Symmetric Spaces

2006-06-01

Armand Borel , Lizhen Ji

Let G be the real locus of a connected semisimple linear algebraic group G defined over Q, and Γ ⊂ G(Q) an arithmetic subgroup. Then the quotient Γ\G is a … Let G be the real locus of a connected semisimple linear algebraic group G defined over Q, and Γ ⊂ G(Q) an arithmetic subgroup. Then the quotient Γ\G is a natural homogeneous space, whose quotient on the right by a maximal compact subgroup K of G gives a locally symmetric space Γ\G/K. This paper constructs several new compactifications of Γ\G. The first two are related to the Borel-Serre compactification and the reductive Borel-Serre compactification of the locally symmetric space Γ\G/K; in fact, they give rise to alternative constructions of these known compactifications. More importantly, the compactifications of Γ\G imply extension to the compactifications of homogeneous bundles on Γ\G/K, and quotients of these compactifications under non-maximal compact subgroups H provide compactifications of period domains Γ\G/H in the theory of variation of Hodge structures. Another compactification of Γ\G is obtained via embedding into the space of closed subgroups of G and is closely related to the constant term of automorhpic forms, in particular Eisenstein series.

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Compactifications of Symmetric and Locally Symmetric Spaces

2006-02-22

Armand Borel , Lizhen Ji

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Compactifications of Symmetric and Locally Symmetric Spaces

2006-01-01

Armand Borel , Lizhen Ji

Armand Borel : a great mathematician of the twentieth century

2006-01-01

Armand Borel

Almost commuting elements in compact Lie groups

2002-01-01

Armand Borel , Robert Friedman , John W. Morgan

1.5.1 for C cyclic . . . . . . . . . . . . . 1.5.1 for C cyclic . . . . . . . . . . . . .

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Compactifications of symmetric and locally symmetric spaces

2002-01-01

Armand Borel , Lizhen Ji

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The Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics (1869-1926)

2001-11-01

Armand Borel , Thomas Hawkins

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Linear algebraic groups in the 19th century

2001-08-07

Armand Borel

Élie Cartan, symmetric spaces and Lie groups

2001-08-07

Armand Borel

Full reducibility and invariants for 𝐒𝐋₂(ℂ)

2001-08-07

Armand Borel

Linear algebraic groups in the 20th century

2001-08-07

Armand Borel

Essays in the History of Lie Groups and Algebraic Groups

2001-08-07

Armand Borel

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Hermann Weyl and Lie groups

2001-08-07

Armand Borel

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The work of Chevalley in Lie groups and algebraic groups

2001-08-07

Armand Borel

Algebraic groups and Galois theory in the work of Ellis R. Kolchin

2001-08-07

Armand Borel

On the Cuspidal Cohomology of S -Arithmetic Subgroups of Reductive Groups over Number Fields

2001-01-01

Armand Borel

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Values of Isotropic Quadratic Forms at S -Integral Points

2001-01-01

Armand Borel

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Cohomology and Spectrum of an Arithmetic Group

2001-01-01

Armand Borel

Values of Quadratic Forms at S -integral points

2001-01-01

Armand Borel

Full Reducibility and Invariants for SL2(C)

2001-01-01

Armand Borel

The Work of Chevalley in Lie Groups and Algebraic Groups

2001-01-01

Armand Borel

Class Functions, Conjugacy Classes and Commutators in Semisimple Lie Groups

2001-01-01

Armand Borel

On Free Subgroups of Semi-simple Groups

2001-01-01

Armand Borel

Values of Zeta-Functions at Integers, Cohomology and Polylogarithms

2001-01-01

Armand Borel

On the Work of E. Cartan on Real Simple Lie Algebras

2001-01-01

Armand Borel

Jean Leray and Algebraic Topology

2001-01-01

Armand Borel

Sous-groupes discrets de groupes p-adiques à covolume borné

2001-01-01

Armand Borel

The School of Mathematics at the Institute for Advanced Study

2001-01-01

Armand Borel

Quelques réflexions sur les mathématiques en général et la théorie des groupes en particulier

2001-01-01

Armand Borel

Algebraic Groups and Galois Theory in the Work of Ellis R. Kolchin

2001-01-01

Armand Borel

Henri Poincaré and Special Relativity

2001-01-01

Armand Borel

Cohomologie d’intersection et L 2-cohomologie de variétés arithmétiques de rang rationnel 2

2001-01-01

Armand Borel

Valeurs de formes quadratiques aux points entiers

2001-01-01

Armand Borel

Values of Indefinite Quadratic Forms at Integral Points and Flows on Spaces of Lattices

2001-01-01

Armand Borel

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A Vanishing Theorem in Relative Lie Algebra Cohomology

2001-01-01

Armand Borel

The Rank Conjecture for Number Fields

2001-01-01

Armand Borel

Sous-groupes épimorphiques de groupes linéaires algéhriques I

2001-01-01

Armand Borel

Sous-groupes épimorphiques de groupes linéaires algéhriques II

2001-01-01

Armand Borel

L 2 -Cohomology and Intersection Cohomology of Certain Arithmetic Varieties

2001-01-01

Armand Borel

Coauthor	Papers Together
Nolan R. Wallach	12
Harish-chandra Harish-Chandra	5
Jean-Pierre Serre	5
Jacques Tits	5
Lizhen Ji	4
Robert Friedman	2
John W. Morgan	2
Gopal Prasad	2
Walter L. Baily	2
T. A. Springer	2
W. Casselman	2
G. D. Mostow	2
Christopher Zeeman	1
Edoardo Vesentini	1
Janós Kollár	1
Jean Siebenthal	1
Albert Schwarz	1
André Haefliger	1
John C. Moore	1
Gregory J. Chaitin	1
Morris W. Hirsch	1
Aldo Andreotti	1
S. MacLane	1
F. Bien	1
F. Bruhat	1
Karen Uhlenbeck	1
Liang Ji	1
Thomas Hawkins	1
C. W. Curtis	1
Jeremy Gray	1
Benoît B. Mandelbrot	1
René Thom	1
J-P. Serre	1
Jun Yang	1
Richard H. Steinberg	1
Rayna J. Carter	1
David Ruelle	1
Raghavan Narasimhan	1
Nagayoshi Iwahori	1
Howard Garland	1
Reinhold Remmert	1
Edward Witten	1
Daniel Friedan	1
James Glimm	1
Michael Atiyah	1
Bertram M. Schreiber	1

Sur La Cohomologie des Espaces Fibres Principaux et des Espaces Homogenes de Groupes de Lie Compacts

1953-01-01

Armand Borel

Some rationality questions on algebraic groups

1957-12-01

Maxwell Rosenlicht

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Homomorphismes "Abstraits" de Groupes Algebriques Simples

1973-05-01

Armand Borel , Jacques Tits

Corners and arithmetic groups

1973-12-01

Armand Borel , J-P. Serre

Some Basic Theorems on Algebraic Groups

1956-04-01

Maxwell Rosenlicht

The subject of algebraic groups has had a rapid development in recent years. Leaving aside the late research by many people on the Albanese and Picard variety, it has received … The subject of algebraic groups has had a rapid development in recent years. Leaving aside the late research by many people on the Albanese and Picard variety, it has received much substance and impetus from the work of Severi on commutative algebraic groups over the complex number field, that of Kolchin, Chevalley, and Borel on algebraic groups of matrices, and especially Weil's research on abelian varieties and algebraic transformation spaces. The main purpose of the present paper is to give a more or less systematic account of a large part of what is now known about general algebraic groups, which may be abelian varieties, algebraic groups of matrices, or actually of neither of these types.

Groupes et algèbres de Lie

1971-01-01

Nicolás Bourbaki

Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes … Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes et algebre de Lie, neuvieme Livre du traite, est consacre aux concepts fondamentaux pour les algebres de Lie. Il comprend les paragraphes: - 1 Definition des algebres de Lie; 2 Algebre enveloppante d une algebre de Lie; 3 Representations; 4 Algebres de Lie nilpotentes; 5 Algebres de Lie resolubles; 6 Algebres de Lie semi-simples; 7 Le theoreme d Ado. Ce volume est une reimpression de l edition de 1971.

Compactification of Arithmetic Quotients of Bounded Symmetric Domains

1966-11-01

Walter L. Baily , Armand Borel

Arithmetic Subgroups of Algebraic Groups

1962-05-01

Armand Borel , Harish-chandra Harish-Chandra

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On Compactifications of the Quotient Spaces for Arithmetically Defined Discontinuous Groups

1960-11-01

Ichirō Satake

In my previous papers [2], [4], I have given a method of compactifying the quotient spaces of the generalized upper half plane with respect to Siegel's modular and, more generally, … In my previous papers [2], [4], I have given a method of compactifying the quotient spaces of the generalized upper half plane with respect to Siegel's modular and, more generally, to any commensurable with Siegel's modular group. Now a similar problem can be considered in a more general situation as follows. Let G be a semi-simple linear algebraic defined over Q, and let G4, GR be the groups consisting of points in G rational over Q, R, respectively. GR is a semisimple Lie with a finite number of connected components. Let K be a maximal compact subgroup of GR and let S = K\GR be the associated (not necessarily connected) symmetric space. Let furthermore Gz be the group of units in G, i.e., the consisting of all elements in G whose coefficients (together with those of its inverse) are rational integers. Then Gz is a discrete subgroup of GR, whose commensurable class is uniquely determined, independently of the choice of the matrix expression of G. Let F be any subgroup of Gq commensurable with Gz. Then one may ask the possibility of constructing a reasonable compactification of the quotient space S/r. To approach this problem, it will be convenient to use a for F, which has been constructed by Weil [7], by means of the reduction theory, in the case where G is a of automorphisms of a semi-simple associative algebra over Q with or without involution. Moreover he has also shown in [6] that these cases cover all semi-simple linear algebraic groups without center, of classical type, over Q, with few exceptions. The purpose of this paper is to give some results on the above problem in the case treated by Weil [7]. The outline of the paper is as follows. We shall recall in ? 1 the main result of our previous paper [3], on which our whole construction will be based, concerning a general method of compactification of a symmetric space S by means of an irreducible, faithful projective representation p of the corresponding Lie group; we give also a trivial generalization of it to the non-connected case. In ? 2 we shall give a general condition (the condition (D)) for a discontinuous r operating on S and for a fundamental set f2 for F, which enables us to construct a suitable compacti-

Linear algebraic groups

1966-01-01

Armand Borel

On quotient varieties and the affine embedding of certain homogeneous spaces

1961-01-01

Maxwell Rosenlicht

We recall that if an algebraic group G operates regularly on a variety V, by a quotient variety is meant a pair (V/G, t), where V/G is a variety and … We recall that if an algebraic group G operates regularly on a variety V, by a quotient variety is meant a pair (V/G, t), where V/G is a variety and t: V-*V/G is a rational map, everywhere defined and surjective, such that two points of V have the same image under t if and only if they have the same orbit on V, and such that, for any xE V, any rational function on V that is G-invariant (i.e., constant on orbits) and defined at x is actually (under the natural injection of function fields Q,(V/G)-*Q(V), ß denoting the universal domain) a rational function on V/G that is defined at tx (cf.[l, exposé 8]).Q,(V/G) must therefore consist precisely of all G-invariant elements of ß(V), so t is separable.A quotient variety need not exist (obvious necessary condition: all orbits on V must be closed), but when it exists it is clearly unique to within an isomorphism; in this case, for any open subset UQV/G, r~lU/G exists and equals U.

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L 2 cohomology of warped products and arithmetic groups

1982-06-01

Steven W. Zucker

On Semi-Simple Automorphisms of Lie Algebras

1955-05-01

Armand Borel , G. D. Mostow

Sur les groupes deLie compacts non connexes

1956-12-01

Jean Siebenthal

Adeles and Algebraic Groups

1982-01-01

André Weil

This volume contains the original lecture notes presented by A. Weil in which the concept of adeles was first introduced, in conjunction with various aspects of C.L. Siegel's work on … This volume contains the original lecture notes presented by A. Weil in which the concept of adeles was first introduced, in conjunction with various aspects of C.L. Siegel's work on quadratic forms.

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Stable Real Cohomology of Arithmetic Groups II

1981-01-01

Armand Borel

Representations of Semisimple Lie Groups VI: Integrable and Square-Integrable Representations

1956-07-01

Harish-chandra Harish-Chandra

On Certain Concepts in the Theory of Algebraic Matric Groups

1948-10-01

E. R. Kolchin

The present paper is a continuation of Chapter I of my previous paper Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations, Ann. of Math., (2) … The present paper is a continuation of Chapter I of my previous paper Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations, Ann. of Math., (2) vol. 49 (1948), pp. 1-42. That paper is referred to below as PV. As in PV, we deal here with algebraic matric groups over an algebraically closed field e of arbitrary characteristic p. Several concepts which were introduced in PV in connection with an algebraic matric group (M, which were discussed there, and which played an important role in the Picard-Vessiot theory as there developed, are here further illuminated. These are the concept of component of the identity of (M, and the concepts of connectedness, solvability, anticompactness, and quasicompactness, properties which 3 may possess. In PV (?6 Theorem 1) it was shown that (M is anticompact (no element of finite order > 1 not divisible by p) if and only if each matrix in @ is reducible to special triangular form (O's below the main diagonal, l's on it). It is shown in ?1 below that this already implies that the whole group (D is reducible to special triangular form (and therefore is solvable). This is not unexpected in the light of classical theorems concerning Lie groups and Lie algebras,' but the present proof has the merit of being purely algebraic (not employing Lie algebras, and being valid for nonzero as well as zero characteristic p). Similarly, in PV (?6 Theorem 2) it was shown that @ is quasicompact (no anticompact algebraic subgroup of order > 1) if and only if each matrix in (D is reducible to diagonal form. It is shown in ?2 below that when 5 is connected this already implies that the whole group (M is reducible to diagonal form (and therefore is abelian). A partial converse of these two results is given in ?3, where it is shown that if @ is abelian then (M = (iq X ha (direct product), where Iq and ha are simultaneously reducible to diagonal form and to special triangular form, respectively. @q and ha are unique. The result of ?4 is mainly for application in ?5, but is perhaps not without interest in itself. It is shown that if 6 is connected, if k is an integer not divisible by p, and if r is a generic point of the underlying manifold of (M, then so is 'rk such a generic point. Thus, the set of matrices in 5 which are not kth powers

Arithmetic of Algebraic Tori

1961-07-01

Takashi Ono

Introduction ? 0. Notation and conventions ? 1. Arbitrary fields 1.1. Duality 1.2. Splitting fields 1.3. Isogenies 1.4. Raising the field of definition 1.5. A structure theorem 1.6. Differential forms … Introduction ? 0. Notation and conventions ? 1. Arbitrary fields 1.1. Duality 1.2. Splitting fields 1.3. Isogenies 1.4. Raising the field of definition 1.5. A structure theorem 1.6. Differential forms ? 2. Local fields 2.1. Maximal compact groups 2.2. Isogenies 2.3. Reduction modulo p ? 3. Fields of dimension 1 3.1. Adelization 3.2. Haar measures 3.3. Canonical correcting factors 3.4. The number p(X, K/k) 3.5. Definition of z(T) 3.6. Isogenies 3.7. Definition of r(a) 3.8. Explicit formula for C(CJA,Ir, (T)k) (resp. c()A, r)) 3.9. Explicit formula for r(cr) 3.10. Main theorem References

Questions of rationality for solvable algebraic groups over nonperfect fields

1963-12-01

Maxwell Rosenlicht

On Algebraic Groups and Homogeneous Spaces

1955-07-01

André Weil

Algebraic Matric Groups and the Picard-Vessiot Theory of Homogeneous Linear Ordinary Differential Equations

1948-01-01

E. R. Kolchin

TABLE OF CONTENTS INTRODUCTION. Historical background. Summary. Notation and terminology. CHAPTER I. ALGEBRAIC MATRIC GROUPS. 1. Reducibility of sets of matrices. 2. Algebraic matric groups. 3. Jordan-Holder-Schreier theorem. 4. Commutator … TABLE OF CONTENTS INTRODUCTION. Historical background. Summary. Notation and terminology. CHAPTER I. ALGEBRAIC MATRIC GROUPS. 1. Reducibility of sets of matrices. 2. Algebraic matric groups. 3. Jordan-Holder-Schreier theorem. 4. Commutator groups. 5. Solvable algebraic matric groups. 6. Anticompact and quasicompact algebraic matric groups. 7. Reducibility to triangular form. 8. Algebraic matric groups with certain types of normal chains. CHAPTER II. SOME P.ESULTS FROM THE THEORY OF ALGEBRAIC DIFFERENTIAL EQUATIONS. 9. Differential rings, fields, and ideals. 10. Differential polynomials. 11. Solutions. 12. Relative isomorphisms. 13. Order. 14. Dependence. 15. Homogeneous linear ordinary differential equations. CHAPTER III. NORMAL DIFFERENTIAL EXTENSION FIELDS. 16. Normal differential extension fields. CHAPTER IV. PICCARD-VESSIOT EXTENSIONS. 17. Picard-Vessiot extensions and their isomorphisms. 18. Normality. 19. Characterization of G. 20. Dimension. 21. Adjunction of new elements. 22. Linear reducibility of L(y). CHAPTER V. LIOUVILLIAN EXTENSIONS. 23. Integrals and exponentials of intgrals. 24. Liouvillian extensions. 25. The principal theorem and some consequences. 26. The proof, first half. 27. The proof, second half. REFERENCES INTRODUCTION

On Some Arithmetic Properties of Linear Algebraic Groups

1959-09-01

Takashi Ono

Let Q be the field of rational numbers and k an extension of Q of finite degree. The multiplicative group k* of k, considered as a group of linear transformations … Let Q be the field of rational numbers and k an extension of Q of finite degree. The multiplicative group k* of k, considered as a group of linear transformations of the vector space k over Q, forms an algebraic group, i.e., a Q-torus in the sense of A. Borel. As is well known in the algebraic number theory, the properties of k* and of its related structures, in particular that of the group Jk of id6les of k, play important roles. On the other hand, let f be a quadratic form on a vector space V over Q of finite dimension. The orthogonal group O( V, f ) composed of all linear transformations of V leaving invariant the form f forms an algebraic group. The properties of the group O( V, f) have essential relations to the arithmetic of the quadratic form f, and the study of these relations has been one of the principal themes in M. Eichler's book Quadratische Formen und Orthogonale Gruppen . Recently the theory of algebraic groups of linear transformations has been systematized by C. Chevalley on the basis of fundamental concepts of algebraic geometry, and the classical mechanism of the Lie theory (correspondence between groups and Lie algebras) has been generalized to the case where the basic field K is an arbitrary field of characteristic 0 (cf., Chevalley [2], [3]). By specializing K to Q, we may apply his methods and results to the study of arithmetic properties of algebraic groups. Thus it could be said that the above two theories, i. e., the arithmetic of k* and that of O( V, f) are two profiles of a kind of unified theory which we might call the arithmetic of algebraic groups. In the present paper, we shall formulate some fundamental concepts for algebraic groups from this point of view and prove some results which might possibly give us some suggestions for further developments in this direction. Thus, in Section 1, we shall introduce the notion of rational characters of algebraic groups and determine the structure of the group of rational characters for a special algebraic group, i. e., a Q-torus (Theorem 1). In Section 2, we shall introduce the notion of G-id6les for an algebraic group G which generalizes the usual notion of id6les of an algebraic number field k and define a subgroup J1(G) of the group J(G) of G-id6les in con266

Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups

2000-01-01

A. Borel , Nolan R. Wallach

The Description for this book, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. (AM-94), will be forthcoming. The Description for this book, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. (AM-94), will be forthcoming.

Kählerian Coset Spaces of Semisimple Lie Groups

1954-12-01

Armand Borel

facts for Fourier expansions on compact group and homogeneous spaces.For summability at points see facts for Fourier expansions on compact group and homogeneous spaces.For summability at points see

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Sur L'Homologie et la Cohomologie des Groupes de Lie Compacts Connexes

1954-04-01

Armand Borel

Geometric Invariant Theory

1982-01-01

David Mumford , John E. Fogarty

Classification of algebraic semisimple groups

1966-01-01

Jacques Tits

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On the Functional Equations Satisfied by Eisenstein Series

1976-01-01

R. P. Langlands

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The Topology of Fibre Bundles. (PMS-14)

1951-12-31

Norman Steenrod

Über eine Beziehung zwischen geschlossenen Lie'schen Gruppen und diskontinuierlichen Bewegungsgruppen euklidischer Räume und ihre Anwendung auf die Aufzählung der einfachen Lie'schen Gruppen

1941-12-01

Eduard Stiefel

Groupes Réductifs Sur Un Corps Local

1972-12-01

François Bruhat , Jacques Tits

Cohomologie des groupes discrets

1971-01-01

Jean-Pierre Serre

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Linear Algebraic Groups

1975-01-01

James E. Humphreys

Compactifications of Symmetric Spaces II: The Cartan Domains

1964-04-01

Calvin C. Moore

On the Hodge theory of Riemannian pseudomanifolds

1980-01-01

Jeff Cheeger

Rationality properties of linear algebraic groups, II

1968-01-01

Armand Borel , T. A. Springer

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La cohomologie mod 2 de certains espaces homogènes

1953-12-01

Armand Borel

Cohomologie Galoisienne

1965-01-01

Jean-Pierre Serre

Théorèmes de finitude en cohomologie galoisienne

1964-12-01

Armand Borel , Jean-Pierre Serre

On some bruhat decomposition and the structure of the hecke rings of p-Adic chevalley groups

1965-12-01

Nagayoshi Iwahori , Hisatake Matsumoto

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A Poisson Formula for Semi-Simple Lie Groups

1963-03-01

Harry Furstenberg

for some bounded function h on the boundary of the disc. The function h(z) determines a function h(g) on G by setting h(g) = h(g(O)). If h(z) is harmonic, it … for some bounded function h on the boundary of the disc. The function h(z) determines a function h(g) on G by setting h(g) = h(g(O)). If h(z) is harmonic, it may be shown that h(g) is annihilated by a certain class of differential operators on G. The Poisson formula (1) may be used to express h(g), and we find that here it takes on a particularly simple form. Namely, if we denote by m the normalized Lebesgue measure on {j z I = 1}, and by gm, the transform of this measure by the group element g E G, then it can be seen that (1) becomes

A realization of Riemannian symmetric spaces

1978-01-01

Toshio Oshima

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Introduction aux groupes arithmétiques

1969-01-01

Armand Borel

Compactifications of symmetric spaces

2007-01-01

Armand Borel , Liang Ji

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Satake and Martin compactifications of symmetric spaces are topological balls

1997-01-01

Lizhen Ji

Symmetric spaces of noncompact type form a very important class of simply connected nonpositively curved Riemannian manifolds with connection to Lie group theory and harmonic analysis. For many applications, it … Symmetric spaces of noncompact type form a very important class of simply connected nonpositively curved Riemannian manifolds with connection to Lie group theory and harmonic analysis. For many applications, it is important to compactify the symmetric spaces. Several compactifications of symmetric spaces have been defined from different points of view. For example, in [17], in order to understand the boundaries which arise in the study of automorphic forms, Satake defined finitely many compactifications of a symmetric space by embedding it into the space of positive definite Hermitian matrices of determinant 1 and then the associated projective space. Another important compactification is the Martin compactification from potential theory. In [10], the Martin compactification of the symmetric space is described in terms of geodesics and the maximal Satake compactification (see Theorem 2.5 below for a precise statement; and see the book [10] for definitions of other compactifications and relations between the various compactifications.) Both the Satake compactifications and the Martin compactification of a symmetric space are topological compactifications. In this note, we prove that they are topologically a closed ball (2.4 and 2.6). In the proof (§4), we use the convexity result of Atiyah [3] that the image of the moment map of a Hamiltonian torus action is a convex polytope (3.1) in order to identify the closure of a flat in the compactifications of the symmetric space (4.1). We also formulate a conjecture on the Martin compactification of simply connected and nonpositively curved Riemannian manifolds (5.1).

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Topology of Lie groups and characteristic classes

1955-01-01

Armand Borel

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Algebraic Groups Over Finite Fields

1956-07-01

Serge Lang

Compactifications of Symmetric Spaces

1964-01-01

Calvin C. Moore

Geometry of compactifications of locally symmetric spaces

2002-01-01

Lizhen Ji , Robert MacPherson

For a locally symmetric space M, we define a compactification M∪M(∞) which we call the "geodesic compactification". It is constructed by adding limit points in M(∞) to certain geodesics in … For a locally symmetric space M, we define a compactification M∪M(∞) which we call the "geodesic compactification". It is constructed by adding limit points in M(∞) to certain geodesics in M. The geodesic compactification arises in other contexts. Two general constructions of Gromov for an ideal boundary of a Riemannian manifold give M(∞) for locally symmetric spaces. Moreover, M(∞) has a natural group theoretic construction using the Tits building. The geodesic compactification plays two fundamental roles in the harmonic analysis of the locally symmetric space:1) it is the minimal Martin compactification for negative eigenvalues of the Laplacian, and 2) it can be used to parameterize the eigenfunctions of the Laplacian in continuous spectrum on L 2 .

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