  In this paper we develop a theory of Besov and Triebel--Lizorkin spaces on
general noncompact Lie groups endowed with a sub-Riemannian structure. Such
spaces are defined by means of hypoelliptic sub-Laplacians with drift, and
endowed with a measure whose density with respect to a right Haar measure is a
continuous positive character of the group. We prove several equivalent
characterizations of their norms, we establish comparison results also
involving Sobolev spaces of recent introduction, and investigate their complex
interpolation and algebra properties.
