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Noninvariant base change identities. (English) Zbl 0868.11026
In this book the author establishes a noninvariant trace formula for \(\mathrm{GL}(n)\) and then gives a new proof of the \(\mathrm{GL}(n)\) base change.
The original proof of this problem was given by J. Arthur and L. Clozel in “Simple algebras, base change, and the advanced theory of the trace formula”, Ann. Math. Studies 120, Princeton Univ. Press (1989; Zbl 0682.10022)]. The approach used by Arthur and Clozel is to get an invariant trace formula from the standard trace formula using certain complicated arguments. That standard trace formula is noninvariant because it is derived from the truncation process. Arthur and Clozel got the base change results by comparing the invariant trace formula and its twisted version.
The author’s new approach is to study the noninvariant trace formula directly, without the trouble of getting into an invariant form. This approach also makes the applications of the trace formula to base change problems easier. Thus the present work can be regarded as a simplification of Arthur and Clozel’s proof. A main difficulty which has been overcome in this book is the proof of a noninvariant fundamental lemma for all functions in unramified Hecke algebras, using a result of Kottwitz on the noninvariant fundamental lemma for unit elements in the unramified Hecke algebras.
The author’s final result is slightly more general than Arthur and Clozel’s. Indeed, because of Mœglin and Waldspurger’s description of the discrete spectrum, it is no longer necessary to restrict oneself to automorphic representations induced from cuspidal ones.

MSC:
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11R39 Langlands-Weil conjectures, nonabelian class field theory
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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