T. A. Springer

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All published works (126)

On symplectic transformations.

2021-12-02

T. A. Springer

This is an English translation, prepared by Wilberd van der Kallen, of the 1951 thesis by T.A. Springer. In this thesis Springer studied the classification of conjugacy classes in the … This is an English translation, prepared by Wilberd van der Kallen, of the 1951 thesis by T.A. Springer. In this thesis Springer studied the classification of conjugacy classes in the symplectic group $Sp_n(K)$, where the commutative field $K$ is of characteristic different from 2. He finds that each conjugacy class is characterized by a system of invariants. These invariants are first of all -- as in the case of the general linear group -- irreducible polynomials and systems of non-negative integers, but secondly also equivalence classes of certain Hermitian forms and of certain quadratic forms. Original Dutch title: Over symplectische transformaties.

Cells for a Hecke Algebra Representation

2019-01-03

T. A. Springer

If $Y$ is an affine symmetric variety for the reductive group $G$ with Weyl group $W$, there exists by Lusztig and Vogan a representation …  If $Y$ is an affine symmetric variety for the reductive group $G$ with Weyl group $W$, there exists by Lusztig and Vogan a representation of the Hecke algebra of $W$ in a module which has a basis indexed by the set $\Lambda$ of pairs $(v, \xi)$, where $v$ is an orbit in $Y$ of a Borel group $B$ and $\xi$ is a $B$-equivariant rank one local system on $v$. We introduce cells in $\Lambda$ and associate with a cell a two-sided cell in $W$.

Some Results on Algebraic Groups with Involutions

2018-06-06

T. A. Springer

IntroductionLet G be a reductive linear algebraic group over the algebraically closed field F of characteristic not 2. Assume given an involution 8 of G, i.e. an automorphism (in the … IntroductionLet G be a reductive linear algebraic group over the algebraically closed field F of characteristic not 2. Assume given an involution 8 of G, i.e. an automorphism (in the sense of algebraic groups) of order 2. Denote by K the fixed point group of 8 and by B a Borel subgroup.Then K haE finitely many orbits in the flag manifold GJB (first proved for F=C in [6]).The geometry of these orbits is of importance in the study of Harish-Chandra modules, as is shown by the results of [11].In the present paper we shall establish a number of basic elementary facts about these orbits or, equivalently, the double cosets BxK.After assembling a number of known results in n° 2, we discuss in n° 3 twisted involutions in Weyl groups.These are needed for the description of double cosets given in n° 4. That section also contains a fairly explicit description of the double cosets as algebraic varieties.N° 5 deals with the open double coset and with those of codimension one.As an application we deduce a result (5.6) about K-fixed vectors in G-modules, which is well-known in characteristic 0 ([4], [12]).Finally, n° 6 contains some information about orbit closures.Similar results have recently been found by Matsuki for F=C [7].The results about double cosets BxK established here bear some refemblance to the familiar results about the Bruhat decomposition into double cosets BxB, but are somewhat more complicated.§ 2. Notations and recollections 2.1.In the sequel, F denotes an algebraically closed field of characteristic not 2. Let G be a linear algebraic group over F, provided with an automorphism e of order 2. Denote by g the Lie algebra of G.The automorphism of g induced by e will also be denoted bye.We shall mainly be interested in the case that G is connected and reductive, but we do not yet assume this.

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Intersection multiplicities and reflection subquotients of unitary reflection groups I

2012-03-24

G Lehrer , T. A. Springer

Remarks on Parabolic Character Sheaves

2011-01-01

T. A. Springer

The paper contains comments on the parabolic character sheaves associated to a connected reductive group G, introduced by Lusztig in [L2].A correspondence is established between parabolic character sheaves and certain … The paper contains comments on the parabolic character sheaves associated to a connected reductive group G, introduced by Lusztig in [L2].A correspondence is established between parabolic character sheaves and certain perverse sheaves on G, equivariant under a subgroup ∆ of G × G, acting on G in the usual way (see Th. 3.8).Section 6 is about comments on results of He in [H3].

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Decompositions related to symmetric varieties

2010-03-27

T. A. Springer

F-groups: General Results

2009-01-01

T. A. Springer

In this chapter the results of the preceding one will be applied to algebraic groups. The notations are as in 2.1. 'Linear algebraic group over F' will be abbreviated to … In this chapter the results of the preceding one will be applied to algebraic groups. The notations are as in 2.1. 'Linear algebraic group over F' will be abbreviated to 'F-group.' As before, F S is a separable closure of F.

Some Algebraic Geometry

2009-01-01

T. A. Springer

This preparatory chapter discusses basic results from algebraic geometry, needed to deal with the elementary theory of algebraic groups. More algebraic geometry will appear as we go along. More delicate … This preparatory chapter discusses basic results from algebraic geometry, needed to deal with the elementary theory of algebraic groups. More algebraic geometry will appear as we go along. More delicate results involving ground fields are deferred to Chapter 11.

Derivations, Differentials, Lie Algebras

2009-01-01

T. A. Springer

We first discuss tangent spaces of algebraic varieties and related algebraic matters. In the second part of the chapters, Lie algebras of linear algebraic groups are introduced and their basic … We first discuss tangent spaces of algebraic varieties and related algebraic matters. In the second part of the chapters, Lie algebras of linear algebraic groups are introduced and their basic properties are established.

Commutative Algebraic Groups

2009-01-01

T. A. Springer

This chapter deals with results about commutative linear algebraic groups which are basic for the theory of the later chapters. The important tori are introduced in 3.2, and we prove … This chapter deals with results about commutative linear algebraic groups which are basic for the theory of the later chapters. The important tori are introduced in 3.2, and we prove the classification theorem 3.4.9 of connected one dimensional groups. The notations are as in the previous chapters.

Topological Properties of Morphisms, Applications

2009-01-01

T. A. Springer

The first part of the chapter deals with general results about morphisms of algebraic varieties. Then these results are applied in the theory of algebraic groups. One of the main … The first part of the chapter deals with general results about morphisms of algebraic varieties. Then these results are applied in the theory of algebraic groups. One of the main items of the chapter is the construction in 5.5 of the quotient of a linear algebraic group by a closed subgroup.

Parabolic Subgroups, Borel Subgroups, Solvable Groups

2009-01-01

T. A. Springer

In this chapter basic ingredients of the theory of linear algebraic groups are introduced: maximal tori, Borel groups, parabolic subgroups. Fundamental results are the conjugacy theorems for Borel groups and … In this chapter basic ingredients of the theory of linear algebraic groups are introduced: maximal tori, Borel groups, parabolic subgroups. Fundamental results are the conjugacy theorems for Borel groups and maximal tori (6.2.7 and6.4.1). The structure theory of connected solvable groups is also treated. The chapter begins with a brief discussion of complete varieties.

Solvable F-groups

2009-01-01

T. A. Springer

This chapter is about solvable groups over a ground field F. The emphasis is on F-split solvable groups and their properties. In 14.4.3 we prove the conjugacy over F of … This chapter is about solvable groups over a ground field F. The emphasis is on F-split solvable groups and their properties. In 14.4.3 we prove the conjugacy over F of two maximal F-tori of a solvable F-group. k, F, … are as before. G is a connected, solvable, linear algebraic group over F. Its unipotent radical - which by 6.3.3 (ii) coincides with the set of its unipotent elements- is denoted by G u .

Reductive F-Groups

2009-01-01

T. A. Springer

This chapter is a continuation of the preceding one. We now consider the case of reductive groups. Some basic results about parabolic subgroups are established. The chapter is mainly devoted … This chapter is a continuation of the preceding one. We now consider the case of reductive groups. Some basic results about parabolic subgroups are established. The chapter is mainly devoted to a discussion of versions over F of the isomorphism and existence theorems. G is a connected, reductive F-group.

Freductive Groups

2009-01-01

T. A. Springer

In this chapter we discuss general F-groups. An important general result is the conjugacy over F of maximal F-split tori (15.2.6). In 15.3 we introduce the root datum of an … In this chapter we discuss general F-groups. An important general result is the conjugacy over F of maximal F-split tori (15.2.6). In 15.3 we introduce the root datum of an F-reductive group. In the case of reductive F-groups the proofs of several results (such as 15.1.3 and 15.3.4) are easier. The notations are as in the previous chapters. G is a connected F-group.

The Existence Theorem

2009-01-01

T. A. Springer

k is an algebraically closed field. This chapter is devoted to the proof of the following existence theorem. k is an algebraically closed field. This chapter is devoted to the proof of the following existence theorem.

More Algebraic Geometry

2009-01-01

T. A. Springer

The next chapters will be devoted to rationality questions in the theory of linear algebraic groups, i.e. questions involving ground fields. The present chapter is preparatory. It discusses basic rationality … The next chapters will be devoted to rationality questions in the theory of linear algebraic groups, i.e. questions involving ground fields. The present chapter is preparatory. It discusses basic rationality results on algebraic varieties.

Linear Algebraic Groups, First Properties

2009-01-01

T. A. Springer

In this chapter algebraic groups are introduced. We establish a number of basic results, which can be handled with the limited amount of algebraic geometry dealt with in the first … In this chapter algebraic groups are introduced. We establish a number of basic results, which can be handled with the limited amount of algebraic geometry dealt with in the first chapter. k is an algebraically closed field and F a subfield. All algebraic varieties are over k.

The Isomorphism Theorem

2009-01-01

T. A. Springer

G denotes a connected, reductive, linear algebraic group over k and T a maximal torus of G. The main result of this chapter is that the root datum ψ (G, … G denotes a connected, reductive, linear algebraic group over k and T a maximal torus of G. The main result of this chapter is that the root datum ψ (G, T) introduced in 7.4.3 determines G up to isomorphism. In the proof of this uniqueness result we shall study in details the way in which G is built up from T and the groups Uα (α ε R) of 8.1.1 (i). We shall get involved with a number of technicalities about root systems.

The exotic nilcone of a symplectic group

2008-09-11

T. A. Springer

Some results on compactifications of semisimple groups

2007-05-15

T. A. Springer

This paper deals with recent results involving a compactification X of a semisimple group G. The emphasis is on the case that G is adjoint and X is its wonderful … This paper deals with recent results involving a compactification X of a semisimple group G. The emphasis is on the case that G is adjoint and X is its wonderful compactification. Group theoretical constructions in G have repercussions in X. The paper describes a number of them.

An extension of Bruhat's lemma

2007-01-26

T. A. Springer

Relèvement de Brauer et représentations paraboliques de GLn(Fq) [d'après G. Lusztig]

2006-11-23

T. A. Springer

Twisted conjugacy in simply connected groups

2006-08-25

T. A. Springer

Some Groups of Type <i>E<sub>7</sub></i>

2006-06-01

T. A. Springer

Abstract An algebraic group of type E 7 over an algebraically closed field has an irreducible representation in a vector space of dimension 56 and is, in fact, the identity … Abstract An algebraic group of type E 7 over an algebraically closed field has an irreducible representation in a vector space of dimension 56 and is, in fact, the identity component of the automorphism group of a quartic form on the space. This paper describes the construction of the quartic form if the characteristic is ≠ 2, 3, taking into account a field of definition F . Certain F -forms of E 7 appear in the automorphism groups of quartic forms over F , as well as forms of E 6 . Many of the results of the paper are known, but are perhaps not easily accessible in the literature.

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Armand Borel: 1923-2003

2004-05-01

James Arthur , Enrico Bombieri , K. Chandrasekharan +5 more

这篇结伴式文章多侧面展现了Armand Borel，他于2003年8月11日去世．8位作者记述他，依次是Serre，Chandrasekharan，Bombieri，Hirzebruch，Springer，Tits，Arthur，Prasad．Borel在代数和拓扑方面的研究足够好，使他在34岁时被聘为IAS(高等研究院)的数学教授．Serre综览他的数学全貌，而Hirzebruch侧重于拓扑方面以不同方式描述他的数学．Springer和Tits讨论Borel在代数群方面的工作，而Arthur评论算术群，以及Borel在这个领域的工作如何奠定自守形式现代理论的基石出．一位研究院的同事说道Borel强烈相信数学的统一性和书面记载的重要性．实现这些信念的方法包括各种角色：编辑、作者、教育家和会议组织者，本文作者中有几位详细描述了这些活动．Borel投入大量的努力到Bourbaki全集的撰稿中，Borel在1998年3月《通报》(Notices of the AMS)里文章“与Nicolas Bourbaki的25年，1949—1973”详细描写了他的经历．广泛认为Bourbaki关于Lie群和Lie代数这一章的写作Borel起了主要作用，这一章已经显出特别持久的价值．Borel是1962—79年间的Annals of Mathematics的编辑，1979—93年间Inventiones Mathematicae的编辑，以及其它杂志的编辑．1998-2000年间他悄悄地以无头衔的副主编方式为《通报》服务，在各种事务上向主编进言，尤其是纪念文章和《通报》与其它国家相应的杂志合作．Borel在策划关于A．Weil的各种文章以及纪念J．Leray和A．Lichnerowicz的文章，起了大而又默默无闻的作用．本文作者中几位描写到Borel著或编的一些书，这些书的列表中共有17本，除了他的文集：论文集【文集】，列在下面方框里．多数书是讨论班的结果，有时与他人合作有时没有．不管讨论班采取何种形式，人们都可相信Borel是每个人的导师．特别的注记是两次美国数学会暑期学校的会议录，1965年在Boulder[3]和1977年在Corvalli 这篇结伴式文章多侧面展现了Armand Borel，他于2003年8月11日去世．8位作者记述他，依次是Serre，Chandrasekharan，Bombieri，Hirzebruch，Springer，Tits，Arthur，Prasad．Borel在代数和拓扑方面的研究足够好，使他在34岁时被聘为IAS(高等研究院)的数学教授．Serre综览他的数学全貌，而Hirzebruch侧重于拓扑方面以不同方式描述他的数学．Springer和Tits讨论Borel在代数群方面的工作，而Arthur评论算术群，以及Borel在这个领域的工作如何奠定自守形式现代理论的基石出．一位研究院的同事说道Borel强烈相信数学的统一性和书面记载的重要性．实现这些信念的方法包括各种角色：编辑、作者、教育家和会议组织者，本文作者中有几位详细描述了这些活动．Borel投入大量的努力到Bourbaki全集的撰稿中，Borel在1998年3月《通报》(Notices of the AMS)里文章“与Nicolas Bourbaki的25年，1949—1973”详细描写了他的经历．广泛认为Bourbaki关于Lie群和Lie代数这一章的写作Borel起了主要作用，这一章已经显出特别持久的价值．Borel是1962—79年间的Annals of Mathematics的编辑，1979—93年间Inventiones Mathematicae的编辑，以及其它杂志的编辑．1998-2000年间他悄悄地以无头衔的副主编方式为《通报》服务，在各种事务上向主编进言，尤其是纪念文章和《通报》与其它国家相应的杂志合作．Borel在策划关于A．Weil的各种文章以及纪念J．Leray和A．Lichnerowicz的文章，起了大而又默默无闻的作用．本文作者中几位描写到Borel著或编的一些书，这些书的列表中共有17本，除了他的文集：论文集【文集】，列在下面方框里．多数书是讨论班的结果，有时与他人合作有时没有．不管讨论班采取何种形式，人们都可相信Borel是每个人的导师．特别的注记是两次美国数学会暑期学校的会议录，1965年在Boulder[3]和1977年在Corvalli

Invariants of a reduced decomposition

2003-02-01

T. A. Springer

Intersection cohomology of B×B-orbit closures in group compactifications

2002-12-01

T. A. Springer

Twisted Composition Algebras

2000-01-01

T. A. Springer , Ferdinand D. Veldkamp

J-algebras and Albert Algebras

2000-01-01

T. A. Springer , Ferdinand D. Veldkamp

Proper J-algebras and Twisted Composition Algebras

2000-01-01

T. A. Springer , Ferdinand D. Veldkamp

Octonions, Jordan Algebras and Exceptional Groups

2000-01-01

T. A. Springer , Ferdinand D. Veldkamp

Triality

2000-01-01

T. A. Springer , Ferdinand D. Veldkamp

Exceptional Groups

2000-01-01

T. A. Springer , Ferdinand D. Veldkamp

Composition Algebras

2000-01-01

T. A. Springer , Ferdinand D. Veldkamp

Cohomological Invariants

2000-01-01

T. A. Springer , Ferdinand D. Veldkamp

The Automorphism Group of an Octonion Algebra

2000-01-01

T. A. Springer , Ferdinand D. Veldkamp

A note concerning fixed points of parabolic subgroups of unitary reflection groups

1999-12-01

G Lehrer , T. A. Springer

Reflection Subquotients of Unitary Reflection Groups

1999-12-01

G Lehrer , T. A. Springer

Abstract Let G be a finite group generated by (pseudo-) reflections in a complex vector space and let g be any linear transformation which normalises G . In an earlier … Abstract Let G be a finite group generated by (pseudo-) reflections in a complex vector space and let g be any linear transformation which normalises G . In an earlier paper, the authors showed how to associate with any maximal eigenspace of an element of the coset gG , a subquotient of G which acts as a reflection group on the eigenspace. In this work, we address the questions of irreducibility and the coexponents of this subquotient, as well as centralisers in G of certain elements of the coset. A criterion is also given in terms of the invariant degrees of G for an integer to be regular for G . A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces.

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Compactification of symmetric varieties

1999-06-01

Corrado De Concini , T. A. Springer

Hecke algebra representations related to spherical varieties

1998-02-11

Jason Mars , T. A. Springer

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a connected reductive group over the algebraic closure of a finite field and let <inline-formula … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a connected reductive group over the algebraic closure of a finite field and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a spherical variety for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We consider perverse sheaves on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which have a weight for the action of a Borel subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and are endowed with an action of Frobenius. This leads to the definition of a “generalized Hecke algebra”, attached to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and of a module over that algebra, attached to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The same algebra and the same module can also be defined using constructible sheaves. Comparison of the two definitions gives, in the case of a symmetric variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivariant sheaves, a geometric proof of results which Lusztig and Vogan obtained by representation theoretic means.

J-structures of Low Degree

1998-01-01

T. A. Springer

Jordan Algebras and Algebraic Groups

1998-01-01

T. A. Springer

Schubert varieties and generalizations

1998-01-01

T. A. Springer

Linear Algebraic Groups

1998-01-01

T. A. Springer

The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. The second edition, thoroughly revised and expanded, extends the theory over arbitrar The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. The second edition, thoroughly revised and expanded, extends the theory over arbitrar

Ideals, the Radical

1998-01-01

T. A. Springer

The Lie Algebras Associated with a J-structure

1998-01-01

T. A. Springer

Relation with Jordan Algebras (Characteristic ≠2)

1998-01-01

T. A. Springer

Simple J-structures

1998-01-01

T. A. Springer

J-structures

1998-01-01

T. A. Springer

Common Coauthors

Coauthor	Papers Together
Ferdinand D. Veldkamp	10
F.D. Veldkamp	3
G Lehrer	3
Richard H. Steinberg	2
R. W. Richardson	2
Armand Borel	2
Eric Bergshoeff	1
W Hermans	1
Johannes A. Lenstra	1
Lauren Stewart	1
H Lennox	1
C A Ramm	1
Jacques Tits	1
信一加藤	1
Jan C. Willems	1
Amanda Bridges	1
Magdalena Fossum	1
Ingrid Meyer	1
Nico H.J. Pijls	1
Edward C. Gruber	1
K Hannan	1
Helen K. Burns	1
A.F. Monna	1
Nagayoshi Iwahori	1
A. Beukers	1
B Krop	1
Joram Lindenstrauss	1
James Truslow Adams	1
Emanuele Isidori	1
Karin Hirsch	1
M N van der Heyde	1
S Rudin	1
Casper G. Schalkwijk	1
B Bos	1
F. van der Blij	1
Diederik F. Janssen	1
Sébastien Wit	1
A Primrose	1
Peter Slodowy	1
William W. Eaton	1
Hanspeter Kraft	1
J. Steenbrink	1
Ju I Manin	1
Leif Strömberg	1
Jack O.L. Saunders	1
Anne Linton	1
Peter J‎. Cameron	1
Jamille Vilas Bôas	1
Mary Beth Fasano	1
Abe Shenitzer	1

Commonly Cited References

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.

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.

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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The Bruhat order on symmetric varieties

1990-01-01

R. W. Richardson , T. A. Springer

Some Results on Algebraic Groups with Involutions

2018-06-06

T. A. Springer

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Regular elements of semisimple algebraic groups

1965-12-01

Robert Steinberg

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LINEAR ALGEBRAIC GROUPS

1970-11-01

B. A. F. Wehrfritz

By A. Borel: pp. x, 398; Cloth $12.50, Paper $4.95. (W. A. Benjamin, Inc., New York, 1969). By A. Borel: pp. x, 398; Cloth $12.50, Paper $4.95. (W. A. Benjamin, Inc., New York, 1969).

Representations of Reductive Groups Over Finite Fields

1976-01-01

Pierre Deligne , G. Lusztig

Linear Algebraic Groups

1998-01-01

T. A. Springer

On a class of jordan algebras

1959-01-01

T. A. Springer

Trigonometric sums, green functions of finite groups and representations of Weyl groups

1976-12-01

T. A. Springer

Sur certains groupes simples

1955-01-01

Catherine Chevalley

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Composition algebras and their automorphisms

1958-01-01

Nathan Jacobson

Oktaven, Ausnahmegruppen und Oktavengeometrie

1985-09-01

Hans Freudenthal

Character sheaves, V

1986-08-01

G. Lusztig

Quadratic forms over fields with a discrete valuation

1955-01-01

T. A. Springer

Questions of rationality for solvable algebraic groups over nonperfect fields

1963-12-01

Maxwell Rosenlicht

Characterization of a Class of Cubic Forms

1962-01-01

T. A. Springer

Orbits, invariants, and representations associated to involutions of reductive groups

1982-06-01

R. W. Richardson

Some arithmetical results on semi-simple lie algebras

1966-12-01

T. A. Springer

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Quadratische Formen und Orthogonale Gruppen

1952-01-01

Martin Eichler

The projective octave plane. I

1960-01-01

T. A. Springer

The 𝐺-stable pieces of the wonderful compactification

2007-02-21

Xuhua He

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a connected, simple algebraic group over an algebraically closed field. There is a partition of … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G overbar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into finite many <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-stable pieces, which was introduced by Lusztig. In this paper, we will investigate the closure of any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-stable piece in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G overbar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We will show that the closure is a disjoint union of some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-stable pieces, which was first conjectured by Lusztig. We will also prove the existence of cellular decomposition if the closure contains finitely many <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-orbits.

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The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group

1959-10-01

Bertram Kostant

On Reduced Exceptional Simple Jordan Algebras

1957-11-01

A. A. Albert , Nathan Jacobson

Linear Algebraic Groups

1991-01-01

Armand Borel

Some groups of transformations defined by Jordan algebras. II. Groups of type F4.

1960-01-01

N. Jacobsen

Article Some groups of transformations defined by Jordan algebras. II. Groups of type F4. was published on January 1, 1960 in the journal Journal für die reine und angewandte Mathematik … Article Some groups of transformations defined by Jordan algebras. II. Groups of type F4. was published on January 1, 1960 in the journal Journal für die reine und angewandte Mathematik (volume 1960, issue 204).

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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The characters of the finite general linear groups

1955-01-01

J. A. Green

Introduction. In this paper we show how to calculate the irreducible characters of the group GL(n, q) of all nonsingular matrices of degree n with coefficients in the finite field … Introduction. In this paper we show how to calculate the irreducible characters of the group GL(n, q) of all nonsingular matrices of degree n with coefficients in the finite field of q elements. These characters have been given for n = 2 by H. Jordan [8], Schur [10], and others, and for n =3 and n =4 by Steinberg [12], who has also [13] done important work in the general case. We are concerned here with characters, that is, characters of representations by matrices with complex coefficients. Let Xi, * * *, XA be the distinct absolutely irreducible ordinary characters of a group 5 of order g. By a of 6 (often called a generalised character or difference character) we mean a class-function 4 on 5 of the form

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Lie Group Representations on Polynomial Rings

1963-07-01

Bertram Kostant

0.Introduction. 1.Let G be a group of linear transformations on a finite dimensional real or complex vector space X.Assume X is completely reducible as a G-module.Let 5 be the ring … 0.Introduction. 1.Let G be a group of linear transformations on a finite dimensional real or complex vector space X.Assume X is completely reducible as a G-module.Let 5 be the ring of all complexvalued polynomials on X, regarded as a G-module in the obvious way, and let JC5 be the subring of all G-invariant polynomials on X.Now let J + be the set of all ƒ £ J having zero constant term and let HQS be any graded subspace such that S=J + S+H is a G-module direct sum.It is then easy to see that

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FINITE GROUPS OF LIE TYPE Conjugacy classes and complex characters

1987-05-01

J. A. Green

By Roger W. Carter: pp. 544. £42.50. (John Wiley & Sons Ltd, 1985) By Roger W. Carter: pp. 544. £42.50. (John Wiley & Sons Ltd, 1985)

Singularities of closures ofK-orbits on flag manifolds

1983-06-01

G. Lusztig , David A. Vogan

The classification of reduced exceptional simple jordan algebras

1960-01-01

T. A. Springer

Composition Algebras and Their Automorphisms

1989-01-01

Nathan Jacobson

The principal objective of the present paper is the study of the automorphisms and groups of automorphisms of composition algebras, that is, the algebras arising from quadratic forms which permit … The principal objective of the present paper is the study of the automorphisms and groups of automorphisms of composition algebras, that is, the algebras arising from quadratic forms which permit composition. These algebras are mainly quaternion algebras and Cayley algebras. The problem of determining the quadratic forms which permit composition (Huryitz's problem) has been treated by many authors (2). In spite of this, there does not appear in any one place a complete solution of this problem in its most general form — which amounts to the determination of the algebras for an arbitrary field and not just to the determination of the possible dimensionalities. We give such a solution here for the case of characteristic not two. Aside from its intrinsic interest and applications to other fields (for example Jordan algebras, absolute valued algebras) we have still another reason for treating the Hurvitz problem again, namely: The analysis of the composition algebras is essential for our study of their automorphisms.

Quelques applications de la cohomologie d'intersection

1982-01-01

T. A. Springer

On Reduced Exceptional Simple Jordan Algebras

1989-01-01

A. A. Albert , Nathan Jacobson

In this paper we shall be concerned only with finite dimensional algebras over an arbitrary field $$ \mathfrak{F} $$ of characteristic not two. Let $$ \mathfrak{A} $$ be an associative … In this paper we shall be concerned only with finite dimensional algebras over an arbitrary field $$ \mathfrak{F} $$ of characteristic not two. Let $$ \mathfrak{A} $$ be an associative algebra over $$ \mathfrak{F} $$ and ab the associative product composition of $$ \mathfrak{A} $$ . Then the vector space $$ \mathfrak{A} $$ is a Jordan algebra $$ {\mathfrak{A}^{\left( + \right)}} $$ relative to the composition a·b = 1/2(ab + ba), that is, this composition satisfies the defining identities (1) $$ a \cdot b = b \cdot a,\,\left[ {\left( {a \cdot a} \right) \cdot } \right] \cdot a = \left( {a \cdot a} \right) \cdot \left( {b \cdot a} \right) $$ The algebra $$ {\mathfrak{A}^{\left( + \right)}} $$ and its subalgebras are called special Jordan algebras.

On the desingularization of the unipotent variety

1976-12-01

Robert Steinberg

Parabolic Character Sheaves, III

2010-01-01

G. Lusztig

The main purpose of this paper is to define a class of simple perverse sheaves (called character sheaves) on certain ind-varieties associated to a loop group.This has applications to a … The main purpose of this paper is to define a class of simple perverse sheaves (called character sheaves) on certain ind-varieties associated to a loop group.This has applications to a geometric construction of certain affine Hecke algebras with unequal parameters (an affine analogue of another construction by the author), as will be shown elsewhere.

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Conjugacy Classes in Algebraic Groups

1974-01-01

Robert Steinberg

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Éléments de géométrie algébrique

1964-01-01

Alexandre Grothendieck

Intersection cohomology of B×B-orbit closures in group compactifications

2002-12-01

T. A. Springer

Reductive Groups are Geometrically Reductive

1975-07-01

William Haboush

Let G be a semi-simple algebraic group over an algebraically closed field, k. Let G act rationally by automorphisms on the finitely generated k-algebra, R. The problem of proving that … Let G be a semi-simple algebraic group over an algebraically closed field, k. Let G act rationally by automorphisms on the finitely generated k-algebra, R. The problem of proving that the ring of invariants, RG, is finitely generated originates with the invariant theorists of the nineteenth century. When k = C, the complex numbers, and G GL (n, C) the question is answered affirmatively by Hilbert's fundamental theorem of invariant theory. The proof involved constructing a G equivariant projection from R to RG and then using it to prove the result algebraically. When k is of characteristic 0 and G is any semi-simple group, by a theorem of H. Weyl, every finite dimensional representation of G is completely reducible. In the 1950's D. Mumford and others (Cartier, Iwahori, Nagata) applied Weyl's theorem to construct a projection from R to RG for any semi-simple group. This made it possible to generalize Hilbert's proof to an arbitrary semi-simple group. Certain geometric applications, particularly to the theory of moduli, made a generalization to groups over fields of positive characteristic highly desirable. In positive characteristic, complete reducibility definitely fails. Hence attempts were made to replace complete reducibility with a weaker condition which would at once hold for all semi-simple groups and make a proof of finite generation of RG possible. The weakest way to state complete reducibility is the following. If V is a finite dimensional G-module containing a G-stable sub-space of co-dimension one, VT, then there is a G-stable line LC V such that V0 E L = V. Mumford conjectured a weaker version of this statement by seeking a complement only in a higher symmetric power of V, SI( V). This is the conjecture as it is stated in the preface to [16]:

Irreducible characters of semisimple lie groups III. Proof of Kazhdan-Lusztig conjecture in the integral case

1983-06-01

David A. Vogan

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.