Stabilization of the twisted trace formula. Vol. 1
The Arthur-Selberg trace formula is an important identity in the spectral geometry of the space \(G(F)\backslash G({\mathbb A})\) for a number field \(F\), where \(\mathbb A\) is the adele ring and \(G\) a reductive linear group over \(F\). It comes about by comparing two ways of computing the trace of an integral operator, spectrally on the one side and geometrically on the other. The trace formula has profound applications in the Langlands theory, where it can be used to show instances of functoriality. The strategy of these proofs is to compare the trace formulae of two different groups, applied to suitable test functions, whose geometric sides of the formula match, so that the ensuing identity of the spectral sides yields the wanted functoriality. In order to match the geometric sides, which are organized by conjugacy classes in the groups, one needs a suitable correspondence between these conjugacy classes. Such a correspondence is much easier obtained if one uses conjugacy over the respective algebraic closures instead of the ground field, i.e., the so called \textit{stable conjugacy classes}. To exploit such a correspondence, one needs a different version of the trace formula, the \textit{stable trace formula}. The stabilization of the trace formula was carried out by \textit{J. Arthur} in three groundbreaking papers in 2001--2003 [Invent. Math. 143, No. 1, 157--220 (2001; Zbl 0978.11025); J. Inst. Math. Jussieu 1, No. 2, 175--277 (2002; Zbl 1040.11038); Ann. Math. (2) 158, No. 3, 769--873 (2003; Zbl 1051.11027)], at the time conditionally based on the fundamental lemma, which in the meantime has been proven \textit{G. Laumon} and \textit{B. C. Ngô} [Ann. Math. (2) 168, No. 2, 477--573 (2008; Zbl 1179.22019)] in the unitary case and in general by \textit{B. C. Ngô} [Publ. Math., Inst. Hautes Étud. Sci. 111, 1--271 (2010; Zbl 1200.22011)]. Langlands functoriality in its full version also incorporates more general assertions than those provable with the stable trace formula. Besides the group \(G({\mathbb A})\), the action of the absolute Galois group \(\Gamma=\mathrm{Gal}(\overline F/F)\) is also incorporated in the general functoriality conjecture. In order to prove more general instances of functoriality, one therefore needs a more general version of the trace formula which also includes Galois twists and which is known as the \textit{twisted trace formula}. Similar to the untwisted case, this formula needs stabilization as well, which is carried out in the sequel of the present book. The volume under review contains five preparatory chapters on the local theory. The first chapter presents the theory of local endoscopy by Kottwitz, Shelstad and Labesse. The chapters II, III and V treat the stabilization of twisted weighted orbital integrals and the intermediary chapter IV treats some results for the special case of the ground field \(\mathbb R\). The present two volume endeavor is an important contribution to the development of Langlands theory. The stabilization of the twisted trace formula will make it possible to show many new cases of functoriality in the future. The breadth of the presentation also makes this book a valuable reference work.
scientific article; zbMATH DE number 6630983