Relative Convolutions. I. Properties and Applications

To study operator algebras with symmetries in a wide sense we introduce a notion of {\em relative convolution operators} induced by a Lie algebra. Relative convolutions recover many important classes of operators, which have been already studied (operators of multiplication, usual group convolutions, two-sided convolution etc.) and their different combinations. Basic properties of relative convolutions are given and a connection with usual convolutions is established. Presented examples show that relative convolutions provide us with a base for systematical applications of harmonic analysis to PDO theory, complex and hypercomplex analysis, coherent states, wavelet transform and quantum theory. KEYWORDS: Lie groups and algebras, convolution operator, representation theory, Heisenberg group, integral representations, Hardy space, Szeg\"o projector, Toeplitz operators, Fock space, Segal--Bargmann space, Bargmann projector, Dirac equation, Clifford analysis, coherent states, wavelet transform, quantization.