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Lectures on Gaussian integral operators and classical groups. (English) Zbl 1211.22001
EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-080-7/pbk). xii, 559 p. (2011).
This book is an elementary self-contained introduction to some constructions of representation theory and related topics of differential geometry and analysis.
The carefully chosen title of this book should alert the reader to a multiplicity of purposes to be served.
The first purpose is to provide an exposition of Gaussian integral operators, i.e., operators of the form
\[ Sf(x) =\int_{\mathbb R^n} \exp\left\{ \frac{1}{2}\sum_{k,l}a_{kl}x_kx_l + \sum_{k,l}b_{kl}x_ky_l + \frac{1}{2}\sum_{k,l}c_{kl}y_ky_l \right\} f(y)\, dy. \]
Such operators appear in analysis, probability theory, and mathematical physics in numerous contexts; the most classical examples are the Fourier transform, the Poisson formula for a solution of the heat equation, and the Mehler formula for the time evolution of a harmonic oscillator.
Beyond these classical integral operators, the author treats “Gaussian operators” in greater generality; for instance, he discusses Gaussian operators in Fock spaces, the Segal-Bargmann transform, the Zak transform, operators with theta-kernels, Gaussian \(p\)-adic operators, the real-adelic correspondence, etc.
In Chapter 1, “Gaussian integral operators”, the author investigates Gaussian integral operators in \(L ^2(\mathbb R^n)\) and discusses the multiplicative structure of Gaussian operators and the boundedness conditions.
The second purpose is to present a non-orthodox introduction to the classical groups. Basically, this topic is covered in Chapters 2–3, “Pseudo-Euclidean geometry and groups U\((p, q)\)”, “Linear symplectic geometry”, which form an independent but highly relevant part of the book; also Chapter 10, “Classical \(p\)-adic groups, introduction”, presents a brief informal introduction to \(p\)-adic numbers and a (not too advanced) discussion of \(p\)-adic classical groups.
Both distinctions and analogies between the real and the \(p\)-adic cases look equally mysterious.
The above two purposes can be pursued independently, but they are not as different as one might think. Gaussian operators are important in the representation theory of infinite-dimensional groups; in a certain sense they replace parabolic induction which is the main tool for the construction of representations of finite-dimensional groups. Infinite-dimensional groups provide an additional point of view to that of classical groups, which produces new phenomena and new problems. The author completely removes infinite-dimensional groups from consideration but leaves Gaussian operators, so this is some kind of view to classical groups from infinity.
On the other hand the detailed analysis of general Gaussian operators is more based on methods of classical groups than on the analytic machinery.
The third purpose is to present an exposition of the “Weil representation”, which is closely related to Gaussian integral operators. Note that from a historical point of view, it is, actually, the “Friedrichs representation” or the “Friedrichs-Segal-Berezin-Shale-Weil” representation; it is interesting that the first four authors were motivated by physics or mathematical physics. The representation theory of real semisimple Lie groups and noncommutative harmonic analysis constituted an important and dynamic branch of mathematics during the 1940–70s. Nowadays, we have an obvious crisis of comprehensibility. As a result, we observe an isolation of that field of study. This is perhaps not so sad, since during the period mentioned above the theory deeply influenced other branches of mathematics (and some of these branches are still dynamic, such as the theory of special functions of several variables, integrable systems, infinite dimensional groups, etc.). It is not clear how to solve this old problem. In this book, the author exposes a piece of the theory that preserves links with numerous branches of pure and applied mathematics and mathematical physics.
The topic and the style of this book are determined by the modern crisis mentioned above. Chapter 4, “The Segal-Bargmann transform”, briefly discusses the Segal-Bargmann transform as a tool of time-frequency analysis and micro-local analysis, and some exotic inversion formulas.
The Segal-Bargmann transform \(\mathcal B\) identifies \(L^2 (\mathbb R^n)\) with the Fock space \(\mathbf F_n\). Gaussian operators in \(\mathbf F_n\) are \(\mathcal B\)-pushforwards of Gaussian operators in \(L ^2(\mathbb R^n)\). It is easy to translate the main results of Chapter 1 to the language of Fock spaces.
However, in Chapter 5, “Gaussian operators in Fock spaces”, the author gives independent proofs.
Chapter 6, “Gaussian operators, Details”, contains more detailed discussion of Gaussian integral operators (spectra, eigenvectors, norms, canonical forms, one-parametric groups and semigroups, explicit expression for matrices, etc.).
The author risks going too deeply in too many directions. As a compromise, he considers here only operators \(\mathbf B[S]: {\mathbf F}_n\to {\mathbf F}_n\).
Chapter 7, “Hilbert spaces of holomorphic functions in matrix balls”, discusses a certain natural one-parametric family of representations of Sp\( (2 n,\mathbb R)\) which includes the Weil representation.
Chapter 8, “The Cartier model”, discusses the realization of the Weil representation of Sp\((2n,\mathbb R) \) in the space of functions on the torus \(\mathbb R^{2n}/\mathbb Z^{2n}\). The construction is rather strange. For instance, the Zak transform \(L^2(\mathbb R^n)\to L^2(\mathbb R^{2n}/\mathbb Z^{2n})\) identifies a space of functions of \(n\) variables and a space of functions of \(2n\) variables.
The author pushes forward the action groups Sp\((2n,\mathbb Z)\) and Sp\((2n ,\mathbb Q)\) to the space of functions on the torus. The basic construction is the explicit formula for the action of the symplectic category \(\mathbf {Sp}\) in the Cartier model.
In Chapter 9, “Gaussian operators over finite fields”, the author constructs a canonical correspondence between Gaussian operators and linear relations over finite fields. This is the main result of the chapter.
Chapter 11, “Weil representation over a \(p\)-adic field”, shows that the category of Gaussian operators over a \(p\)-adic field is equivalent to the Nazarov category. The adelic version of the construction is also briefly discussed. At the end, the author considers a funny “integral” operator connecting functions of the real and of the \(p\)-adic variables.
There are numerous problems of varying levels of difficulty in the book.
The book is addressed to graduate students and researchers in representation theory, differential geometry, and operator theory. The reader is supposed to be familiar with standard university courses in linear algebra, functional analysis, and complex analysis. Some familiarity with Lie groups and Lie algebras would also be useful.

MSC:
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
22E50 Representations of Lie and linear algebraic groups over local fields
47B50 Linear operators on spaces with an indefinite metric
47G10 Integral operators
53C35 Differential geometry of symmetric spaces
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
22E46 Semisimple Lie groups and their representations
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