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Let be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell-Hausdorff multiplication. This allows us to define a generalized multiplication $fg={\left(f^{\vee }\ast g^{\vee }\right)}^{\wedge }$ of two functions in the Schwartz class (*), where ${}^{\vee }$ and ${}^{\wedge }$ are the Abelian Fourier transforms on the Lie algebra and on the dual * and ∗ is the convolution on the group . In the operator analysis on nilpotent Lie groups an important notion is the one of symbolic calculus which can be viewed as a higher order generalization...