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 EMS Series of Lectures in Mathematics 2011; 571 pp; softcover Volume: 13 ISBN-10: 3-03719-080-9 ISBN-13: 978-3-03719-080-7 List Price: US$78 Member Price: US$62.40 Order Code: EMSSERLEC/13 This book is an elementary self-contained introduction to some constructions of representation theory and related topics of differential geometry and analysis. Topics covered include the theory of various Fourier-like integral operators such as Segal-Bargmann transforms, Gaussian integral operators in $$L^2$$ and in the Fock space, integral operators with theta-kernels, the geometry of real and $$p$$-adic classical groups and symmetric spaces. The heart of the book is the Weil representation of the symplectic group (real and complex realizations, relations with theta-functions and modular forms, $$p$$-adic and adelic constructions) and representations in Hilbert spaces of holomorphic functions of several complex variables. This book is addressed to graduate students and researchers in representation theory, differential geometry, and operator theory. Prerequisites are standard university courses in linear algebra, functional analysis, and complex analysis. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in representation theory, differential geometry, and operator theory. Table of Contents Gaussian integral operators Pseudo-Euclidean geometry and groups $$\mathrm{U}(p, q)$$ Linear symplectic geometry The Segal-Bargmann transform Gaussian operators in Fock spaces Gaussian operators. Details Hilbert spaces of holomorphic functions in matrix balls The Cartier model Gaussian operators over finite fields Classical $$p$$-adic groups. Introduction Weil representation over a $$p$$-adic field Addendum Bibliography Notation Index