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The base change fundamental lemma for central elements in parahoric Hecke algebras. (English) Zbl 1194.22019
Let $$E/F$$ be finite unramified extension of $$p$$-adic fields, and let $$\theta$$ be a generator of $$\mathrm{Gal}(E/F)$$. Let $$G$$ be an unramified connected reductive group over $$F$$. The article under review proves the fundamental lemma for parahoric Hecke algebras in the case of stable base change from $$G(F)$$ to $$G(E)$$. Let $$J_F$$ be a parahoric subgroup of $$G(F)$$ and let $$J_E$$ be the corresponding parahoric subgroup of $$G(E)$$. Let $$Z(\mathcal H_{J_F}(G(F)))$$ and $$Z(\mathcal H_{J_E}(G(E)))$$ be the centers of the Hecke algebras corresponding to these parahorics. The author defines a “base change homomorphism”
$b:Z(\mathcal H_{J_E}(G(E))) \longrightarrow Z(\mathcal H_{J_F}(G(F)))$
and proves that if $$\phi\in Z(\mathcal H_{J_E}(G(E)))$$, then $$\phi$$ and $$b(\phi)$$ are associated in the sense that the stable orbital integral of $$b(\phi)$$ over a semisimple orbit in $$G(F)$$ agrees with the stable $$\theta$$-twisted orbital integral of $$\phi$$ over a corresponding orbit in $$G(E)$$. This generalizes results of L. Clozel [Duke Math. J. 61, No. 1, 255–302 (1990; Zbl 0731.22011)] and J.-P. Labesse [Duke Math. J. 61, No. 2, 519–530 (1990; Zbl 0731.22012)] in the case of spherical Hecke algebras. The present version of the fundamental lemma can be applied to compute the local factors of Hasse-Weil zeta functions associated to certain Shimura varieties with parahoric level structure at a given prime $$p$$.
The proof that $$\phi$$ and $$b(\phi)$$ are associated proceeds by induction on the semisimple rank of $$G$$. There are two main steps, as in the spherical setting. First, via the induction hypothesis, descent formulas, and other reasoning, the author reduces to the case of strongly regular elliptic semisimple orbits in an adjoint group $$G$$. In the second part, he proves the statement in this more restricted case. This uses a global argument, akin to that of Clozel, which involves the twisted trace formula of P. Deligne, D. Kazhdan and M.-F. Vignéras [Représentations des groupes réductifs sur un corps local (Hermann, Paris), 33–117 (1984; Zbl 0583.22009)], as well as the stabilization of its geometric side due to Kottwitz.
The author also gives a short discussion of the relationship between the article under review and other recent work, including that of Ngô, Laumon, and Waldspurger.

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 20G25 Linear algebraic groups over local fields and their integers
##### Keywords:
base change; fundamental lemma; parahoric; Hecke algebra
Full Text:
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