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Conditional base change for unitary groups. (English) Zbl 1071.22025

In the theory of automorphic forms Langlands has not only formulated a philosophy as to the relationships between automorphic forms in several contexts but he has also proposed that the natural way to prove these is by means of comparison using the Selberg Trace Formula. In treating base change one has to study the role of the Galois group and one uses correspondingly a twisted Selberg Trace Formula. This formula combines analytic and arithmetic aspects; to understand this completely involves a process called “stabilization” of the Trace Formula; the technique is known as “endoscopy”. The problem is to show that the “generic unit”-elements of the Hecke algebra behave as they should. This has proved to be very difficult in general, despite a number of successes. One alternative approach to a complete solution is to try to obtain partial results which, while not satisfying maximalist dreams, suffice for the applications in mind. The purpose of this paper is to do precisely this in comparing unitary groups (associated with a quadratic extension of number fields) with the corresponding general linear groups. The point is that by a suitable choice of test function one can reduce the discussion to the cuspidal part of the spectrum and this suffices for a number of significant applications. The argument uses a large number of advances made by the authors separately and by L. Clozel in several papers.
The results are a little too technical to be given in detail here but the structure is the following. Let \(\pi\) be an irreducible cuspidal automorphic representation of a group of type \(U_n\) such that one or two of the local components satisfy some additional condition. (This can be replaced or augmented by the condition that the derived group is anisotropic.) Then the representation constructed from the local correspondence (here called, not quite appropriately, the Jacquet-Langlands transfer) is automorphic and cuspidal. The local correspondence is defined by an identity of characters (up to sign) on elliptic elements. It is also possible to characterize partially the image of the global “Jacquet-Langlands” transfer. The theory applied to the Archimedean places also allows one to discuss the property of being cohomological under transfer.

MSC:

22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F75 Cohomology of arithmetic groups
20G25 Linear algebraic groups over local fields and their integers
20G30 Linear algebraic groups over global fields and their integers
20G35 Linear algebraic groups over adèles and other rings and schemes
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