Some L2 properties of semigroups of measures on Lie groups

We investigate the induced action of convolution semigroups of probability measures on Lie groups on the L 2-space of Haar measure. Necessary and sufficient conditions are given for the infinitesimal generator to be self-adjoint and the associated symmetric Dirichlet form is constructed. We show that the generated Markov semigroup is trace-class if and only if the measures have a square-integrable density. Two examples are studied in some depth where the spectrum can be explicitly computed, these being the n-torus and Riemannian symmetric pairs of compact type.