In the representation theory of reductive p-adic groups, a lot of general structure was developed by Bernstein: supercuspidal representations, cuspidal support, Bernstein components and so on.

Let G be the group SL(2) over a finite extension F of Q_p, p odd. For a fixed r ≥ 0, we identify the elements of the Bernstein center of G supported in the Moy-Prasad G-domain G_{r^+}, by characterizing them spectrally. We study the behavior of convolution with such elements on orbital integrals of functions in C^∞_c(G(F)), proving results in the spirit of semisimple descent. These are ‘depth r versions’ of results proved for general reductive groups by J.-F. Dat, R. Bezrukavnikov, A. Braverman and D. Kazhdan.