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The twisted trace formula according to the Friday Morning Seminar. (La formule des traces tordue d’après le Friday Morning Seminar.) (French) Zbl 1272.11070
CRM Monograph Series 31. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9441-5/hbk). xxvi, 234 p. (2013).
The twisted trace formula is a generalization for the Arthur-Selberg trace formula. It calculates the truncated trace of the operators \(\rho(f) \circ \theta\) on a connected reductive group \(G\) over a number field for suitable test functions \(f\) on \(G(\mathbb{A})\), where \(\theta\) is (induced by) an automorphism of finite order of \(G\). Taking \(\theta = \mathrm{id}\) reduces to the non-twisted case. Moreover, one can also introduce an automorphic character \(\omega\) in this picture. It is used in the study of cyclic base change, automorphic induction, and the endoscopic classification for representations of classical groups, just to mention a few.
The “Friday Morning Seminar” at the the Institute for Advanced Study in Princeton during 1983–1984 was aimed at providing a solid base for the non-invariant twisted trace formula, such as the one used by Y. Flicker [The trace formula and base change for \(\text{GL}(3)\). Berlin etc.: Springer-Verlag (1982; Zbl 0481.10023)]. According to Langlands’ preface, the lectures given by Clozel, Labesse and Langlands were written hastily. In view of its numerous applications, the lack of a firm ground for the twisted trace formula became a serious issue.
Thanks to the work of Labesse and Waldspurger under review, a rigorous treatment is finally available. Their text is divided into four parts. Part 1 sets up the combinatorial apparatus, reduction theory and Labesse’s language of twisted spaces. Part 2 and 3 are concerned with the truncated kernel, from which one obtains the so-called coarse trace formula; a crucial role is played by the “basic identity” in section 8.2. The most difficult part is the Part 4 concerning the refined spectral side. The method proposed in the Morning Seminar requires certain estimates for intertwining operators and their derivatives, whereas the authors uses a technique of Waldspurger to circumvent such difficulties.
It is hard to overestimate the importance of this work. Nevertheless, the non-invariant twisted trace formula itself is not sufficient. For example, the works of Arthur-Mok require the stable twisted trace formula which is even deeper. The work in progress of Mœglin and Waldspurger on stabilization will hopefully put all these theories on a firm ground.

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings