Compactifications of Symmetric and Locally Symmetric Spaces

Borel, Armand; Ji, Lizhen

doi:10.1007/0-8176-4430-x_2

Cited by 76 publications

(153 citation statements)

References 31 publications

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“…This yields the grouptheoretic compactification C gp (M ). This turns out to be equivariantly isomorphic to the maximal Satake and Furstenberg compactifications (see [19], [5]). In the case of buildings, since points with the same support have identical stabilisers, this approach cannot offer better than a compactification of the set Res sph (X), which generalizes the approach of [20] in which only the set of vertices is compactified.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Combinatorial and group-theoretic compactifications of buildings

Caprace¹,

Lécureux²

2011

Ann. inst. Fourier

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Section: Annales De L'institut Fouriermentioning

confidence: 99%

Combinatorial and group-theoretic compactifications of buildings

Caprace¹,

Lécureux²

2011

Ann. inst. Fourier

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“…This follows from Lemma 6. For part (2), recall that exactness of a sequence of sheafs is equivalent to exactness of the corresponding sequence of stalks at each point. This allows us to reduce (2) to Lemma 5.…”

Section: For Objects V W Of Mod G We Use the Notationmentioning

confidence: 99%

On Emerton’s p-adic Banach Spaces

Hill¹

2010

International Mathematics Research Notices

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Abstract. The purpose of the current paper is to introduce some new methods for studying the p-adic Banach spaces introduced by Emerton [9]. We first relate these spaces to more familiar sheaf cohomology groups. As an application, we obtain a more general version of Emerton's spectral sequence. We also calculate the spaces in some easy cases. As a consequence, we obtain a number of vanishing theorems. 1.1.Cohomology of arithmetic quotients. Let G be a linear algebraic group over a number field k. We choose a maximal compact subgroup K ∞ ⊂ G(k ∞ ) and let K • ∞ be the identity component in K ∞ . This paper is concerned with the cohomology of the following "arithmetic quotients":Fix once and for all a finite prime p of k and let p be the rational prime below p. By a "tame level" we shall mean a compact open subgroup K p of G(A p f ). For a field E of characteristic zero, the level K p Hecke algebra is defined by H(K p , E) = {f : K p \G(A p f )/K p → E : f has compact support}. Given a finite dimensional algebraic representation W of G over an extension E/k p , one defines a local system V W on each arithmetic quotient Y (K f ). We define the classical cohomology groups of tame level K p as follows: where K p ranges over the compact open subgroups of G(k p ). The symbol " * " denotes either the empty symbol, meaning usual cohomology, or "c", meaning compactly supported cohomology. There is a smooth action of G(k p ) on H • cl. (K p , W ), and there is also an action of the level K p Hecke algebra H(K p , E). The systems of Hecke eigenvalues arising in H • cl., * (K p , W ) are of considerable interest in number theory. Traditionally, the classical cohomology groups have been studied using the theory of automorphic representations (for example in [4]). Recently, Emerton introduced a new method for studying the classical cohomology groups. Instead of studying the space of automorphic forms on G, Emerton introduced a collection of p-adic Banach spaces, from which the classical cohomology can be recovered. Emerton's spaces are defined as follows:For a finite extension E/Q p we also definẽhas the structure of a Banach space over E, where the unit ball is defined to be the O E -span of the image ofH • * (K p , Z p ). It is also convenient to consider the direct limit of these groups over the tame levels:We have the following actions on these spaces:(1) The group G(A p f ) acts smoothly onH • * (G, Z p ); the subspaceH • * (K p , E) may be recovered as the K p -invariants inH • * (G, E). (2) The Hecke algebra H(K p , E) acts onH • * (K p , E). (3) The group G(k p ) acts continuously, but not usually smoothly on the spacesH • * (K p , −). (4) For a finite extension E/k p , we defineH • * (K p , E) kp−loc.an. to be the subspace of k p -locally analytic vectors inH • (K p , E) (see [10]). The Lie algebra g of G acts on the subspaceH • * (K p , E) kp−loc.an. . Emerton proved (Theorem 2.2.11 of [9]) that his spaces are related to the classical cohomology groups by the following spectral sequence of smoothThe spectral sequenc...

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“…We will give examples of wonderful compactifications in Section 6. For good references on compactifications of symmetric spaces, see [BJ06,O78,Sa60] or Chapter 4 in [S84].…”

Section: Examplesmentioning

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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Compactifications of Symmetric and Locally Symmetric Spaces

Cited by 76 publications

References 31 publications

Combinatorial and group-theoretic compactifications of buildings

Combinatorial and group-theoretic compactifications of buildings

On Emerton’s p-adic Banach Spaces

Book Review: Homogeneous spaces and equivariant embeddings

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