The fundamental lemma for stable base change. (English) Zbl 0731.22011

Clozel, Laurent

Duke Math. J. 61, No. 1, 255-302 (1990).

Let F be a p-adic field of characteristic zero and \(O_ F\) its ring of integers. Let G be an unramified connected reductive group over F arisen by base extension from a smooth reductive group scheme over \(O_ F\). Let E/F be an unramified extension of degree l and \(\sigma\) a generator of Gal(E/F). We have the base change map b: \({\mathcal H}_ E\to {\mathcal H}_ F\) between Hecke algebras of functions on G(E) and G(F) invariant under \(G(O_ E)\) and \(G(O_ F)\), respectively. There is a norm map \({\mathcal N}\) sending elements in G(E) to stable conjugacy classes of elements in G(F) and, for a semisimple element of G(F), its stable orbital integral is defined.
The main result of this paper is: Theorem. Assume that stable orbital integrals are compatibly normalized. Let \(\phi\in {\mathcal H}_ E\) and \(f=b\phi \in {\mathcal H}_ F\). (i) Assume \(\delta\in G(E)\) and \(\gamma ={\mathcal N}\delta\) is a regular stable conjugacy class in G(F). Then \(\Phi_ f^{st}(\gamma)=\Phi^{st}_{\phi,\sigma}(\delta)\). (ii) Assume \(\gamma\in G(F)\) is regular, and is not in the image of the norm map. Then \(\Phi_ f^{st}(\gamma)=0\). A finer result for semisimple \(\gamma\) of the above theorem is also given.

Reviewer: K.I.Ohta (Tokyo)

MSC:

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

Keywords:

p-adic field; connected reductive group; base extension; smooth reductive group scheme; base change; Hecke algebras; norm map; stable conjugacy classes; semisimple element; stable orbital integral

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