## Stable trace formula: Elliptic singular terms.(English)Zbl 0577.10028

The duality theorems for the Galois cohomology of tori over a number field F are needed in the stabilization of the elliptic regular terms in the trace formula for a connected reductive group G over F. These terms are indexed by elliptic regular elements of G(F), and their connected centralizers are the tori to which the duality theory is applied. The connected centralizer of an elliptic singular element is a connected reductive group, and the stabilization of the corresponding terms in the trace formula will require an analog of the duality theory valid for all connected reductive groups over F. This paper develops such a theory and sketches its application to the stabilization of the trace formula.

### MSC:

 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11R34 Galois cohomology 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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