Singularités et transfert

Author's comments (2021-06-22): The paper as presented here is not the published paper. That was unfortunately modified, namely slightly abridged, by the editors without consulting the author and without his approval. The present paper, the original paper, is the preferred form.

Author's comments: This text is provisional from a mathematical point of view, but it may be some time before the obstacles described in the concluding sections are overcome. Serious progress has been made by Ali Altuğ.

It has been easy to misconstrue the principal purpose of this paper and of the previous paper, at least my principal purpose. It was to introduce the use of the Poisson formula in combination with the stable transfer as a central tool in the development of the stable trace formula and its applications to global functoriality. Unfortunately the review in Math. Reviews was inadequate, simply reproducing the abtract, written not by me but by the editors, ``A transfer similar to that for endoscopy is introduced in the context of stably invariant harmonic analysis on reductive groups. For the group \(\mathrm{SL}(2)\), the existence of the transfer is verified and some aspects of the passage from the trace formula to the Poisson formula are examined.'' This transfer is for me a central issue for harmonic analysis on reductive groups over local fields. The problems it raises have, so far as I know, not been solved even over \(\mathbf R\) and \(\mathbf C\). Its construction for \(\mathrm{SL}(2)\) over \(p\)-adic fields, \(p\) odd, was, and remains, for me an interesting application of the explicit formulas of Sally-Shalika for the characters of that group.