  In this article, we give a general construction of spectral triples from
certain Lie group actions on unital C*-algebras. If the group G is compact and
the action is ergodic, we actually obtain a real and finitely summable spectral
triple which satisfies the first order condition of Connes' axioms. This
provides a link between the "algebraic" existence of ergodic action and the
"analytic" finite summability property of the unbounded selfadjoint operator.
More generally, for compact G we carefully establish that our (symmetric)
unbounded operator is essentially selfadjoint. Our results are illustrated by a
host of examples - including noncommutative tori and quantum Heisenberg
manifolds.
