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On local character relations. (English) Zbl 0923.11081

A fundamental problem in the Langlands program is to stabilize the trace formula. One purpose of the present paper is to lay some local foundations for the future study of this problem. The ultimate goal is to express the invariant distributions in the trace formula explicitly in terms of stable distributions on endoscopic groups. Here endoscopic groups \(G'(F)\) are certain quasi-split groups which are associated to, but typically less complicated than a reductive group \(G\) over a \(p\)-adic field \(F\).
Another purpose of this paper is to study a family of transfer maps from functions on \(G(F)\) to functions on endoscopic groups \(G'(F)\). Some new properties of transfer maps are deduced under the assumption of the fundamental lemma on both a group and its Lie algebra. This amounts to establishing certain identities between characters on \(G(F)\) and stable characters on \(G'(F)\).
The two main theorems are given in section 6, p. 549.
Reviewer: Y.Ye (Iowa City)

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
22E50 Representations of Lie and linear algebraic groups over local fields
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