Semigroups of measures in non-commutative harmonic analysis

We find the invariant convolution semigroups of measures on the Heisenberg group which are analogous to the familiar semigroups in the abelian context from modern potential theory, and we show that the abelian theory may be obtained as the limit of the Heisenberg case as Planck's constant tends to zero. Work supported in part by the NSF.