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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.

##### MSC:
 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
##### Keywords:
Arthur-Selberg trace formula; twisted spaces