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Part of the book series: Modern Birkhäuser Classics ((MBC))

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Abstract

This chapter begins with a presentation of the adèles of the rational numbers. This locally compact ring A = ∏′ ℚ p is a restricted product of all possible completions of ℚ — the field of rational numbers. When G is a real or p-adic algebraic group and Γ a congruence subgroup Strong Approximation results enable us to embed L 2(Γ\G) into L 2(G(ℚ)\G(A)) (see Section 6.3 for a precise formulation). The adèles in general and the spectral decomposition of L 2(G(ℚ)\G(A)) in particular give a convenient way to state results on «all» spaces L 2(Γ\G) together. This way we will see, for example, that the Selberg conjecture is a special case of a more general conjecture which asserts that no complementary series representations appear as local factors of subrepresentation of L 2(PGL(ℚ)\PGL 2(A)). Another special case of this general conjecture is a theorem of Deligne (see (6.2.2)). This theorem, which is, in fact, a representation theoretic reformulation of Ramanujan Conjecture (known also as Petersson’s conjecture), is extremely important to us. This is the theorem which is responsible for both the final solution of the Banach-Ruziewicz problem and the construction of Ramanujan graphs. (To be precise we should say that both problems can be solved using some weaker results proved earlier by Rankin and Eichler, respectively, but Deligne’s Theorem gives a unified approach as well as better results in the Ruziewicz problem (see Chapter 9).)

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© 1994 Springer Basel AG

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Lubotzky, A. (1994). Spectral Decomposition of L 2(G(ℚ)\G(A)). In: Discrete Groups, Expanding Graphs and Invariant Measures. Modern Birkhäuser Classics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0346-0332-4_6

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  • DOI: https://doi.org/10.1007/978-3-0346-0332-4_6

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0346-0331-7

  • Online ISBN: 978-3-0346-0332-4

  • eBook Packages: Springer Book Archive

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