In his influential work from the 1970s, Marc Rieffel explained how unitary induction can be neatly framed within C∗-algebras by using Hilbert modules and the concept of Morita equivalence. In the last several years, Pierre Clare has begun to study parabolic induction, which is the mainstay of Harish-Chandra-style representation theory, from the same C∗-algebraic point of view. I shall introduce Pierre’s basic construction, and then consider the problem of framing “parabolic restriction” within C∗-algebra theory. For tempered representations — roughly speaking, for the reduced rather than the full group C∗-algebra — one can expect adjunction relations between parabolic induction and restriction. The investigation of these relations leads to some interesting asymptotic geometry — for SL(2,ℝ) this is the geometry of the wave equation on the hyperbolic plane. An as-yet poorly understood issue is that the constructions involve the smooth structure of the tempered dual, as captured by a smooth subalgebra, and not just the topology, as captured by the reduced C∗-algebra.