Plongements d’espaces homogènes

Morales, Domingo; Vust, Th.

doi:10.1007/bf02564633

Cited by 219 publications

(179 citation statements)

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“…This compactification is very elementary, in particular when G is the automorphism group of a nondegenerate bilinear form. It shares some common features with the so-called wonderful compactifications of algebraic groups over an algebraically closed field constructed by De Concini and Procesi [DCP83] or Luna and Vust [LV83], as well as with the compactifications constructed by Neretin [Ner98,Ner03]. After completing this note, we learned that this compactification had first been discovered by He [He02]; we still include our original self-contained description for the reader's convenience.…”

Section: Introductionmentioning

confidence: 69%

Compactification of certain Clifford-Klein forms of reductive homogeneous spaces

Guéritaud¹,

Guichard²,

Kassel³

et al. 2017

Michigan Math. J.

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Abstract. We describe smooth compactifications of certain families of reductive homogeneous spaces such as group manifolds for classical Lie groups, or pseudo-Riemannian analogues of real hyperbolic spaces and their complex and quaternionic counterparts. We deduce compactifications of Clifford-Klein forms of these homogeneous spaces, namely quotients by discrete groups Γ acting properly discontinuously, in the case that Γ is word hyperbolic and acts via an Anosov representation. In particular, these Clifford-Klein forms are topologically tame.

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Section: Introductionmentioning

confidence: 69%

Compactification of certain Clifford-Klein forms of reductive homogeneous spaces

Guéritaud¹,

Guichard²,

Kassel³

et al. 2017

Michigan Math. J.

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“…In Chapter 3 the important Luna-Vust theory of embeddings [LV83] is discussed. The chapter includes an extension of this theory by D. A. Timashev [T97].…”

Section: The Bookmentioning

confidence: 99%

Book Review: Homogeneous spaces and equivariant embeddings

Ólafsson¹

2013

Bull. Amer. Math. Soc.

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Basic conceptsGroup actions on manifolds, algebraic varieties and other sets, and geometric objects have played an important role in geometry, analysis, representation theory, and physics for a long time. The book by D. A. Timashev is a welcome survey of old and new results on actions of algebraic groups on algebraic varieties. Basic facts on algebraic groups, homogeneous spaces, and equivariant embeddings are presented. Then the discussion concentrates more on a detailed description of special and interesting classes where deeper results can be obtained. Those cases include symmetric spaces, weakly symmetric spaces, spherical varieties, spaces of lower rank and complexity, and so-called wonderful varieties. Classification of several categories of homogeneous spaces are given. The book is about the algebraic side of group actions, but to complement the book, we take a more analytic viewpoint.Let us start by recalling some basic concepts. Let G be a group, and let X be a set. A G-action on X is a map GˆX Ñ X, often written as pa, xq Þ Ñ a¨x " a pxq, such that a Þ Ñ a is a group homomorphism from G into the group of bijections on X. If a G-action on X is given, then X is said to be a G-set. For a fixed x P X the map a Þ Ñ a¨x is the orbit map and G¨x is a G-orbit. The subgroup G x " ta P G | a¨x " xu is the stabilizer of x in G. We say that X is homogeneous if X " G¨x for some point, x P X. In that case X " G¨x for all points x in X. If X and Y are two G-sets, then a map ϕ : X Ñ Y is G-equivariant, or a G-map, if ϕpa¨xq " a¨ϕpxq for all a P G and all x P X. If X is homogeneous, thenIf G is a Lie group acting smoothly on a manifold X (always assumed separable), then G x is closed for all x P X and G{G x is a manifold. The group G acts smoothly on G{G x and G{G x is isomorphic to X as a manifold and as a G-space. In the algebraic category, the variety G{G x » X if the map aG x Þ Ñ a¨x o is separable. To explain the title of the book we can now say that G{H ϕ ãÑ X is an equivariant embedding if X is a normal variety with a G-action and ϕ is G-equivariant map with an open image in X. In that case it is common to identify G{H with ϕpG{Hq Ă X. Finally, if τ : G Ñ G

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“…Methods here strongly rely on the theory of algebraic groups. We do not describe any of these in detail, but refer the reader to [1,4,11]. Now suppose G/H is a complex homogeneous space; i.e., G is a complex analytic Lie group and H is a closed, complex subgroup.…”

Section: Book Reviewsmentioning

confidence: 99%

Book Review: Lie group actions in complex analysis

Gilligan¹

1997

Bull. Amer. Math. Soc.

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Plongements d’espaces homogènes

Cited by 219 publications

References 12 publications

Compactification of certain Clifford-Klein forms of reductive homogeneous spaces

Compactification of certain Clifford-Klein forms of reductive homogeneous spaces

Book Review: Homogeneous spaces and equivariant embeddings

Book Review: Lie group actions in complex analysis

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