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Introduction to the Schwartz space of \(\Gamma\) \(\backslash G\). (English) Zbl 0723.22010

Let G be the group of \({\mathbb{R}}\)-rational points on a reductive algebraic group defined over \({\mathbb{Q}}\) and \(\Gamma\) an arithmetic subgroup of G. One of the fundamental results in harmonic analysis on \(\Gamma\setminus G\) due to Langlands is that \(L^ 2(\Gamma \setminus G)\) has a decomposition parametrized by the \(\Gamma\)-association classes of \({\mathbb{Q}}\)-parabolic subgroups of G (called a parabolic decomposition of \(L^ 2(\Gamma \setminus G))\) [R. P. Langlands, On the functional equations satisfied by Eisenstein series (Lect. Notes Math. 544, 1976; Zbl 0332.10018); cf. also Harish-Chandra, Automorphic forms on semisimple Lie groups (Lect. Notes Math. 62, 1968; Zbl 0186.047)]. The aim of the paper under review is to rebuild the analogous theory on a new kind of function space \({\mathcal S}(\Gamma \setminus G)\), called the Schwartz space of \(\Gamma\setminus G\), and to apply it to the cohomology theory of \(\Gamma\).
First, the author introduces the space \({\mathcal S}(\Gamma \setminus G)\), as the space of functions on \(\Gamma\setminus G\) satisfying a certain decreasing condition with respect to an appropriate norm on \(\Gamma\setminus G\). It is shown that (1) \({\mathcal S}(\Gamma \setminus G)\) becomes a nuclear Fréchet space and gives a smooth representation of G of moderate growth, and (2) the Z(\({\mathfrak g})\)-finite, K-finite distributions of \(\Gamma\setminus G\) which extend to continuous functionals on \({\mathcal S}(\Gamma \setminus G)\) are automorphic forms on \(\Gamma\setminus G\), where Z(\({\mathfrak g})\) is the center of the universal enveloping algebra of the complexification of G and K is a maximal compact subgroup of G.
Next, the author defines cuspidal Schwartz functions and Eisenstein series maps, and studies a duality relation between the constant term maps and the Eisenstein series maps. Similarly to the \(L^ 2\)-case, these objects are essential in building a parabolic decomposition of \({\mathcal S}(\Gamma \setminus G)\). The principal result is the following: Set \({\mathcal S}_{{\mathcal P}}(\Gamma \setminus G)={\mathcal S}(\Gamma \setminus G)\cap L^ 2_{{\mathcal P}}(\Gamma \setminus G)\) for each \(\Gamma\)- association class \({\mathcal P}\). Then, the inclusion \({\mathcal S}_{{\mathcal P}}(\Gamma \setminus G)\hookrightarrow {\mathcal S}(\Gamma \setminus G)\) induces a topological isomorphism \({\mathcal S}(\Gamma \setminus G)\cong \oplus {\mathcal S}_{{\mathcal P}}(\Gamma \setminus G)\), where \({\mathcal P}\) ranges over the \(\Gamma\)-association classes.
Finally, it is shown that the above decomposition of \({\mathcal S}(\Gamma \setminus G)\) implies the corresponding direct sum decomposition of the cohomology of \(\Gamma\).
The first part of the paper contains a self-contained exposition of some basic notions and techniques which were hitherto used by various authors without sufficient explanations.

MSC:

22E40 Discrete subgroups of Lie groups
11F70 Representation-theoretic methods; automorphic representations over local and global fields
57T10 Homology and cohomology of Lie groups
11F75 Cohomology of arithmetic groups
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A80 Analysis on other specific Lie groups
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