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).)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Basel AG
About this chapter
Cite this chapter
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
Download citation
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