  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.
