Harmonic analysis on semisimple \(p\)-adic Lie algebras
Many questions in harmonic analysis on reductive groups can be transferred to give analogous questions on Lie algebras. In particular, this holds for the comparison of orbital integrals, a problem which lies in the center of the most fruitful applications of the trace formula. As in the group case there are in general some unproven conjectures the most important of which seems to be the fundamental lemma which is the conjectured equality of stable and relative orbital integrals evaluated at the characteristic functions of hyperspecial Lie subalgebras of the Lie algebra of the initial group \(G\) and some endoscopic group \(H\). As is shown in [\textit{J.-L. Waldspurger}, Compos. Math. 105, No. 2, 153-236 (1997; Zbl 0871.22005)] the fundamental lemma would already imply the transfer conjecture which allows to compute general orbital integrals on \(G\) in terms of orbital integrals on \(H\) thus allowing induction arguments in comparison of orbital integrals of two different groups \(G\) and \(G'\). The present paper gives a well written overview of recent work on these conjectures. Especially a formula is given for computing the dimension of the space of stably invariant distributions in the space spanned by the real components of a nilpotent orbital integral.
scientific article; zbMATH DE number 1184314