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Zeta integrals, Schwartz spaces and local functional equations. (English) Zbl 1425.11001
Lecture Notes in Mathematics 2228. Cham: Springer (ISBN 978-3-030-01287-8/pbk; 978-3-030-01288-5/ebook). viii, 139 p. (2018).
The author described the main features of his book as follows: “According to Sakellaridis, many zeta integrals in the theory of automorphic forms can be produced or explained by appropriate choices of a Schwartz space of test functions on a spherical homogeneous space, which are in turn dictated by the geometry of affine spherical embeddings. We pursue this perspective by developing a local counterpart and try to explicate the functional equations. These constructions are also related to the $$L^2$$-spectral decomposition of spherical homogeneous spaces in view of the Gelfand-Kostyuchenko method. To justify this viewpoint, we prove the convergence of $$p$$-adic local zeta integrals under certain premises, work out the case of prehomogeneous vector spaces and re-derive a large portion of Godement-Jacquet theory. Furthermore, we explain the doubling method and show that it fits into the paradigm of $$L$$-monoids developed by L. Lafforgue, B. C. Ngô and others, by reviewing the constructions of Braverman and Kazhdan. In the global setting, we give certain speculations about global zeta integrals, Poisson formulas and their relation to period integrals”.
The book consists of 8 chapters, namely: 1) Introduction containing a review of prior works, a description of main objects and notation; 2) Geometric Background (spherical varieties, Cartan decompositions, geometric data); 3) Analytic Background (integration, Gelfand-Kostyuchenko method); 4) Schwartz Spaces and Zeta Integrals (coefficients of smooth representations, the local functional equation, connection with $$L^2$$ theory); 5) Convergence of Some Zeta Integrals (cellular decompositions, smooth asymptotics); 6) Prehomogeneous Vector Spaces (Fourier transform of half-densities, local Godement-Jacquet integrals); 7) The Doubling Method (doubling zeta integrals, relation to reductive monoids); 8) Speculation on the Global Integrals (theta distributions, relation to periods, global functional equation and Poisson formula).

##### MSC:
 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11M41 Other Dirichlet series and zeta functions 11S40 Zeta functions and $$L$$-functions 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11S90 Prehomogeneous vector spaces 22E50 Representations of Lie and linear algebraic groups over local fields
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