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

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)
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References:

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