The fourfold way in real analysis. An alternative to the metaplectic representation
In order to create a sound foundation for the theory of theta series, A. Weil constructed in 1964 the metaplectic representation of the double cover \(\tilde { Sp}(n,{\mathbb R})\) of the symplectic group \(Sp(n,{\mathbb R})\). Of central importance in this theory is the Gaussian function \(e^{-\pi x^2}\). In the book under review, the author establishes a representation, called the \textit{anaplectic representation}, of a certain finite cover of \(Sp(n,{\mathbb R})\). The role of the Gaussian is played by the Bessel function \(| x| ^{1/2} I_{-1/4}(\pi x^2)\), which lies in the kernel of the harmonic oscillator. As this function grows exponentially, the usual \(L^2\)-inner product needs to be replaced by an indefinite hermitian pairing. The space of the representation is a space of holomorphic functions which satisfy growth conditions in the four directions \(+\infty, -\infty, +i\infty, -i\infty\), thus giving rise to the name of the book. As in the metaplectic case, the image of the anaplectic representation contains a ``Fourier transform'', this one being adapted to the new ``inner product''. The development of anaplectic analysis in the whole rather uses elementary real algebraic geometry than Hilbert space methods. In the first chapter of the book the one-dimensional anaplectic representation is introduced, the \(n\)-dimensional in the second. The third chapter is aimed at the development of a symbolic calculus for the anaplectic analysis, and in the fourth and last chapter, the anaplectic representation is embedded in a continuous family of more general representations. The author expresses his wish that this new theory may turn out useful in quantum mechanics, partial differential equations, and in special functions theory.
scientific article; zbMATH DE number 5015521