Computations with Bernstein projectors of SL(2)

For the p-adic group G = SL(2), we present results of the computations of the sums of the Bernstein projectors of a given depth. Motivation for the computations is based on a conversation with Roger Howe in August 2013. The computations are elementary, but they provide an expansion of the delta distribution \(\delta _{1_{G}}\) into an infinite sum of G-invariant locally integrable essentially compact distributions supported on the set of topologically unipotent elements. When these distributions are transferred, by the exponential map, to the Lie algebra, they give G-invariant distributions supported on the set of topologically nilpotent elements, whose Fourier transforms turn out to be characteristic functions of very natural G-domains. The computations in particular rely on the SL(2) discrete series character tables computed by Sally–Shalika in 1968. This new phenomenon for general rank has also been independently noticed in recent work of Bezrukavnikov, Kazhdan, and Varshavsky.