An example of Fourier transforms of orbital integrals and their endoscopic transfer (Q1383488)

Orbital integrals occur in the ``geometric side'' of Selberg's trace formula. Let \(F\) be a \(p\)-adic field. The paper under review computes the Fourier transform of a regular orbital integral on the Lie algebra of \(SL_2(F)\) as an integral of regular orbital integrals with explicit coefficients. In the last two sections the result is translated into the language of \textit{J.-L. Waldspurger} [J. Reine Angew. Math. 465, 41-99 (1995; Zbl 0829.11030) and Compos. Math. 105, 153-236 (1997; Zbl 0871.22005)]. These last papers obtain ``endoscopic transfer'' for \(SL_n\). ``Endoscopic transfer tries to relate instable orbital integrals of a group and stable orbital integrals of so-called endoscopic groups''. ``For \(G=SL_2\) the non-trivial endoscopic groups are one-dimensional algebraic tori defined by quadratic extensions \(E| F\) and a stable orbital integral in such a torus is just evaluation at an element''.