Harmonic analysis on semisimple \(p\)-adic Lie algebras
type: Item
instance of: scholarly article
title: Harmonic analysis on semisimple \(p\)-adic Lie algebras
zbMATH Open document ID: 0899.22009
author: Robert Kottwitz
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/224792
full work available at URL: Kottwitz.MAN.html
review text: 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.
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 1184314
zbMATH Keywords: orbital integrals
zbMATH Keywords: \(p\)-adic Lie algebra
zbMATH Keywords: harmonic analysis
zbMATH Keywords: reductive groups
zbMATH Keywords: trace formula
zbMATH Keywords: fundamental lemma
zbMATH Keywords: hyperspecial Lie subalgebras
reviewed by: Anton Deitmar
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1184314