Endomorphisms of linear algebraic groups

Steinberg, Richard H.

doi:10.1090/memo/0080

Cited by 570 publications

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“…An automorphism ψ of an algebraic group G is called semisimple if the derivative dψ is a semisimple linear map on its Lie algebra (see [St1] Section 7, pages 27-28). A crucial fact needed in Steinberg's paper [St], is his well-known result on the existence of a (semisimple) automorphism-invariant maximal torus in an algebraic group over an algebraically closed field (Theorem 7.5, [St1]).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…A crucial fact needed in Steinberg's paper [St], is his well-known result on the existence of a (semisimple) automorphism-invariant maximal torus in an algebraic group over an algebraically closed field (Theorem 7.5, [St1]). In order to prove results on the power maps of real algebraic groups, as done in Section 5, we require an analogue of the above result for such groups and in the following theorem we show that this holds in a much more general context, namely, for arbitrary algebraic groups defined over fields of characteristic zero.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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Automorphism invariant Cartan subgroups and power maps of disconnected groups

Chatterjee

2010

Math. Z.

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Abstract. We extend a well-known result of R. Steinberg on the existence of an invariant maximal torus under a semisimple automorphism of an algebraic group over an algebraically closed field. We show that the same result holds when the underlying field is of characteristic zero, but not necessarily algebraically closed. We next study surjectivity of the power maps g → g n of disconnected algebraic groups of characteristic zero. In the case of disconnected real algebraic groups we apply our generalisation of Steinberg's result to obtain results on the surjectivity of the power maps. We also extend a result of A. Borel on weak exponentiality in real Lie groups by relating it with the surjectivity of the square map.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Automorphism invariant Cartan subgroups and power maps of disconnected groups

Chatterjee

2010

Math. Z.

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“…is generated by its unipotent elements [St,Theorem 12.4], each of the form exp(x) for some nilpotent matrix x. LetS + denote the algebraic subgroup of GL k generated by the corresponding φ x (A 1 ).…”

Section: The Main Theoremmentioning

confidence: 99%

Type A images of Galois representations and maximality

Hui

Larsen

2016

Math. Z.

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“…One can use any W -invariant lattice in R n and Kato works in this more general situation. When the one uses the weight lattice P , a result of Steinberg [St,4.2,5.3] says that the stabilizer W t of a point t ∈ T under the action of W is always a reflection group. Because of this Kato's criterion takes a simpler form.…”

Section: Principal Series Modulesmentioning

confidence: 99%

Representations of Rank Two Affine Hecke Algebras

Ram

2003

Advances in Algebra and Geometry

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This paper classifies and constructs explicitly all the irreducible representations of affine Hecke algebras of rank two root systems. The methods used to obtain this classification are primarily combinatorial and are, for the most part, an application of the methods used in [Ra1]. I have made special effort to describe how the classification here relates to the classifications by Langlands parameters (coming from p-adic group theory) and by indexing triples (coming from a q-analogue of the Springer correspondence). There are several reasons for doing the details of this classification: (a) The proof of the one of the main results of [Ra1] I single out Jacqui Ramagge with special thanks for everything she has done to help with this project: from the most mundane typing and picture drawing to deep intense mathematical conversations which helped to sort out many pieces of this theory. Her immense contribution is evident in that some of the papers in this series on representations of affine Hecke algebras are joint papers.A portion of this research was done during a semester long stay at Mathematical Sciences Research Institute where I was supported by a Postdoctoral Fellowship. I thank MSRI and National Science Foundation for support of my research. Definitions and preliminary resultsThe Weyl group.Let R be a reduced irreducible root system in R n , fix a set of positive roots R + and let {α 1 , . . . , α n } be the corresponding simple roots in R. Let W be the Weyl group corresponding to R. Let s i denote the simple reflection in W corresponding to the simple root α i and recall that W can be presented by generators s 1 , s 2 , . . . , s n and relationsThe Iwahori-Hecke algebra. Fix q ∈ C * such that q is not a root of unity. The Iwahori-Hecke algebra H is the associative algebra over C defined by generators T 1 , T 2 , . . . , T n and relationswhere m ij are the same as in the presentation of W . For w ∈ W define T w = T i 1 · · · T i p where s i 1 · · · s i p = w is a reduced expression for w. By [Bou, Ch. IV §2 Ex. 23], the element T w does not depend on the choice of the reduced expression. The algebra H has dimension |W | and the set {T w } w∈W is a basis of H.The group X. The fundamental weights are the elements ω 1 , . . . , ω n of R n given by ω i , α ∨ j = δ ij . The weight lattice is the W -invariant lattice in R n given byLet X be the abelian group P except written multiplicatively. In other words, X = {X λ | λ ∈ P }, and X λ X µ = X λ+µ = X µ X λ , for λ, µ ∈ P . Affine Hecke algebras 3Let C[X] denote the group algebra of X. There is a W -action on X given by wX λ = X wλ for w ∈ W , X λ ∈ X, which we extend linearly to a W -action on C[X].The affine Hecke algebra. The affine Hecke algebraH associated to R and P is the algebra given byH = C-span{T w X λ | w ∈ W, X λ ∈ X} where the multiplication of the T w is as in the Iwahori-Hecke algebra H, the multiplication of the X λ is as in C[X] and we impose the relationThis formulation of the definition ofH is due to Lusztig [Lu2] following work of Bernstein...

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Endomorphisms of linear algebraic groups

Cited by 570 publications

References 0 publications

Automorphism invariant Cartan subgroups and power maps of disconnected groups

Automorphism invariant Cartan subgroups and power maps of disconnected groups

Type A images of Galois representations and maximality

Representations of Rank Two Affine Hecke Algebras

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