  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.
