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Galois properties of cohomological automorphic forms on GL$$(n)$$. (English) Zbl 1017.11026
From the introduction: Let $$F$$ be a $$p$$-adic field. The author’s article [Invent. Math. 129, 75-119 (1997; Zbl 0886.11029), cited as [H1]], combined with the arguments of C. Bushnell, G. Henniart and P. Kutzko [Ann. Sci. Éc. Norm. Supér. (4) 31, 537-560 (1998; Zbl 0915.11055)], leads to a construction for a correspondence between supercuspidal representations of $$\text{GL} (n,F)$$ and irreducible $$n$$-dimensional representations of the Weil group $$W_F$$ of $$F$$ on the $$\ell$$-adic cohomology of a certain family of rigid analytic spaces, introduced by Drinfel’d. The construction was global and as such included a determination of the local Galois representations attached to automorphic forms on certain kinds of unitary groups. These unitary groups $$GG$$ are associated to division algebras with involution over CM fields $${\mathcal K}$$ containing imaginary quadratic fields $${\mathcal K}_0$$. Under the hypotheses of the author’s article (loc. cit.), $$p$$ splits in $${\mathcal K}_0$$ and the local field $$F$$ occurs as the completion of $${\mathcal K}$$ at a $$p$$-adic place $$v$$. The local representation of $$\text{Gal} (\overline{F}/F)$$ occurs in a compatible system of $$\lambda$$-adic representations of $$\text{Gal} (\overline{\mathcal K}/{\mathcal K})$$, defined on the $$\ell$$-adic cohomology of a Shimura variety $$\overline{S}$$ attached to $$GG$$. Thus the author’s construction (loc. cit.) determines the action of the decomposition group at $$v$$ on the part of the cohomology of $$\overline{S}$$ corresponding to representations that are supercuspidal at $$v$$.
The main purpose of the present article is to extend this result to $$\ell$$-adic cohomology of $$\overline{S}$$ with coefficients in $$\ell$$-adic local systems defined by finite-dimensional algebraic representations of $$GG$$. Theorem 1.7 of the author’s [Invent. Math. 134, 177-210 (1998; Zbl 0921.11060), cited as [H2]] asserts that the author’s correspondence of [H1] extends to this more general setting. This assertion is used there in [H2] to prove that the correspondence [H1] preserves $$\varepsilon$$-factors of pairs, provided the global and local correspondences on the cohomology of $$\overline{S}$$ are compatible at non-supercuspidal places as well.
The proof of Theorem 1.7 of [H2] is given in §5 of the present paper. An independent proof of this theorem is given in M. Harris and R. Taylor [The geometry and cohomology of some simple Shimura varieties, Ann. Math. Stud. 151, Princeton Univ. Press (2001; Zbl 1036.11027)]. The correspondence of [H1] is based on a Hochschild-Serre spectral sequence (Lemma 6) for the $$\ell$$-adic cohomology of $$p$$-adically uniformized Shimura varieties. This was proved using unpublished results of Berkovich, who studied the action of $$G$$ on the cohomology of analytic spaces $$X$$ with $$G$$-action, where $$G$$ is a $$p$$-adic analytic group. Given a torsion sheaf $$F$$ on $$X$$ with smooth $$G$$-action, with orders prime to $$p$$, the unpublished notes [V. G. Berkovich, Étale equivariant sheaves on $$p$$-adic analytic spaces, manuscript in preparation; and private communications April 1995] construct acyclic resolutions of $$F$$ with smooth $$G$$-action. We apply Berkovich’s construction and show in §2 that the resolutions of the constant sheaves $$\mathbb{Z}/ \ell^m\mathbb{Z}$$ satisfy a Mittag-Leffler condition as $$m$$ varies. This allows us to extend Lemma 6 of [H1] to torsion sheaves and twisted $$\ell$$-adic coefficients. The construction for torsion sheaves becomes somewhat more complicated when $$\ell$$ is not prime to the orders of finite subquotients of $$\text{GL} (n,F)$$ (the “non-banal” case, in the terminology of M.-F. Vignéras [Représentations $$\ell$$-modulaires d’un groupe réductif $$p$$-adique avec $$\ell\neq p$$, Birkhäuser, Boston (1996; Zbl 0859.22001)]).
The final section studies the $$\ell$$-torsion in the cohomology of the Drinfel’d rigid spaces when $$\ell$$ is banal. Some of the results of the present paper were announced at the 1996 Durham conference on arithmetic and Galois representations. This article belongs to the author’s series aiming at a proof of the local Langlands conjecture. This has now been achieved by the author and R. Taylor and also by G. Henniart [see the overview by J. Rogawski, Notices Am. Math. Soc. 47, 35-41 (2000; Zbl 1009.11063)].

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11S37 Langlands-Weil conjectures, nonabelian class field theory 22E50 Representations of Lie and linear algebraic groups over local fields
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