Harmonic analysis on semisimple \(p\)-adic Lie algebras. (English) Zbl 0899.22009

Kottwitz, Robert E.

Doc. Math., Extra Vol. ICM Berlin 1998, vol. II, 553-562 (1998).

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 [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.

Reviewer: A.Deitmar (Heidelberg)

MSC:

22E35	Analysis on \(p\)-adic Lie groups
22E50	Representations of Lie and linear algebraic groups over local fields
11F85	\(p\)-adic theory, local fields
11F70	Representation-theoretic methods; automorphic representations over local and global fields

Keywords:

orbital integrals; \(p\)-adic Lie algebra; harmonic analysis; reductive groups; trace formula; fundamental lemma; hyperspecial Lie subalgebras

Citations:

Zbl 0871.22005