Computations with Bernstein projectors of SL(2)

TL;DR

This paper computes sums of Bernstein projectors for SL(2) over p-adic fields, revealing a new expansion of the delta distribution into G-invariant distributions supported on topologically unipotent elements, with implications for harmonic analysis.

Contribution

It provides explicit elementary computations of Bernstein projectors for SL(2), expanding the delta distribution into a sum of invariant distributions supported on topologically unipotent elements.

Findings

Expansion of delta distribution into invariant distributions

Distributions supported on topologically unipotent elements

Connections to Fourier transforms and G-domains

Abstract

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.