Lectures on Gaussian Integral Operators and Classical Groups

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 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.