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Automorphic forms on reductive groups. (English) Zbl 1137.11032

Sarnak, Peter (ed.) et al., Automorphic forms and applications. Papers of the IAS/Park City Mathematics Institute, Park City, UT, USA, July 1–20, 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-2873-1/hbk). IAS/Park City Mathematics Series 12, 7-39 (2007).
This paper is mainly an extended survey (with selected concise proofs) of the basic theory of automorphic forms on reductive groups, up to and including the convergence of Eisenstein series for sufficiently large values of the parameters. The organization follows basically the special case \(\text{SL}_2(\mathbb R)\) as expounded in the author’s well-known book [see A. Borel, Automorphic forms on \(\text{SL}_2(\mathbb R)\). Cambridge: Cambridge University Press (1997; Zbl 0912.11023)].
The paper starts with the notion and first properties of automorphic forms and a review of reductive groups. Motivation is drawn from the examples \(\text{GL}_n(\mathbb R)\), \(\text{SL}_n(\mathbb R)\) and the orthogonal groups defined by quadratic forms. The focus is on split tori, roots, parabolic subgroups, the Langlands decomposition…This is followed by a review of arithmetic groups, reduction theory and Siegel sets (“representing the cusps”). The latter sets are used to reformulate the growth condition satisfied by automorphic forms in various equivalent ways. Of crucial importance is the notion of constant term of a continuous function on \(\Gamma\setminus G\) which leads to the concept of cusp form. The so-called “basic estimate” for the difference between \(f\) and its constant term is used to prove Harish-Chandra’s theorem on the finiteness of the dimension of spaces of automorphic forms. For any \(\alpha\in C^\infty_c(G)\) the convolution with \(\alpha\) is a Hilbert-Schmidt-operator on \({^0 L^2}(\Gamma\setminus G)\) (Theorem of Gelfand and Piatetski-Shapiro). This yields that, as a \(G\)-module, \({^0L^2}(\Gamma\setminus G)\) is a Hilbert direct sum of countably many irreducible \(G\)-invariant closed subspaces with finite multiplicities. The following two sections deal with automorphic forms, the regular representation and a decomposition of the space of automorphic forms culminating in the formulation of a theorem of Langlands. The section on growth estimates gives the necessary preparations for the proof of convergence of certain Eisenstein series in Section 12. The latter theorem is given here a formulation and proof (due to J. Bernstein) which apparently cannot be found elsewhere in this form in the literature.
This paper appears to be the swan song written in the admirable impeccable style of the well-known author, whose original investigations and expository writing have contributed so much to the advancement of the science of mathematics and whose death (in 2003) is an irreparable loss to it.
For the entire collection see [Zbl 1116.11003].

MSC:

11F55 Other groups and their modular and automorphic forms (several variables)
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
22E30 Analysis on real and complex Lie groups
22E40 Discrete subgroups of Lie groups
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods

Citations:

Zbl 0912.11023
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