Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups

Borel, Armand; Wallach, Nolan R.

doi:10.1090/surv/067

Cited by 685 publications

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“…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.…”

Section: 1mentioning

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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Section: 1mentioning

confidence: 99%

On Emerton’s p-adic Banach Spaces

Hill¹

2010

International Mathematics Research Notices

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“…If we let τ = η + ξ/2, then τ ∈ 1 2 O + (because we may assume η i and η i + ξ i are positive), and because |ξ i | T 1+ we have τ i + T 2 max(η i , η i + ξ i ) for all i. Therefore by Deligne's bound,…”

Section: Lemma 12 For φ a Fixed Hecke-maass Cusp Form Or Pure Incompmentioning

confidence: 98%

“…Then F k may be thought of as a harmonic 1-form which represents a cohomology class in H 1 (Y, V k ) (this is why π is referred to as being of cohomological type). However, we will not use this point of view in this paper and shall only refer the reader to the book of Borel and Wallach [1] where correspondences of this kind are described in detail. We wish to establish the equidistribution of the probability measures |F k | 2 dv on Y , in generalisation of holomorphic QUE over Q.…”

Section: Automorphic Formsmentioning

confidence: 99%

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Mass equidistribution for automorphic forms of cohomological type on 𝐺𝐿₂

Marshall

2011

J. Amer. Math. Soc.

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We extend Holowinsky and Soundararajan’s proof of quantum unique ergodicity for holomorphic Hecke modular forms on S L ( 2 , Z ) SL(2,\mathbb {Z}) , by establishing it for automorphic forms of cohomological type on G L 2 GL_2 over an arbitrary number field which satisfy the Ramanujan bounds. In particular, we have unconditional theorems over totally real and imaginary quadratic fields. In the totally real case we show that our result implies the equidistribution of the zero divisors of holomorphic Hecke modular forms, generalising a result of Rudnick over Q \mathbb {Q} .

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“…H • (n, π K )) the corresponding Lie algebra cohomology (resp. homology) (see [3], [4]). We have the following isomorphism of AM -modules (see [20], p57):…”

Section: The Selberg Zeta Functionmentioning

confidence: 99%