Authors’ summary: “We aim to show, using the example of a Riemannian symmetric pair \((G,K)=(SL_2(\mathbb{R}),SO(2))\), how contraction ideas may be applied to functional calculi constructed on coadjoint orbits of Lie groups. We construct such calculi on principal series orbits and generic orbits of the Cartan motion group \(V\ltimes K\), and show how the two are related. Since the calculi are adapted to the representations traditionally attached to the orbits, we recover at the Lie algebra level the contraction results of A. H. Dooley and J. W. Rice [Trans. Am. Math. Soc. 289, 185-202 (1985; Zbl 0566.22015)]”.