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Supercuspidal representations in the cohomology of Drinfel’d upper half spaces; elaboration of Carayol’s program. (English) Zbl 0886.11029

Let \(F\) be a finite extension of \(\mathbb Q_p\), and let \(\Sigma_N\) be an inverse system of rigid analytic étale Galois coverings of the nonarchimedean upper half space \(\Omega \subset \mathbb P^{n-1}_F\) indexed by a positive integer \(N\). These coverings are equivariant for the action of \(GL(n,F)\) and for the group \(B^\times\) of units in a division algebra \(B\) over \(F\) with invariant \(1/n\). It was conjectured by H. Carayol [Automorphic forms, Shimura varieties, and \(L\)-functions, Vol. II, Perspect. Math. 11, 15-40 (1990; Zbl 0704.11049)] that the direct limit of the cohomology of the \(\Sigma_N\) decomposed as the sum, over square-integrable representations \(\pi\) of \(GL(n,F)\), of \(\pi \otimes JL(\pi) \otimes \widetilde{\sigma} (\pi)\), where \(JL(\pi)\) is the representation of \(B^\times\) associated to \(\pi\) by the generalized Jacquet-Langlands correspondence, and where \(\widetilde{\sigma} (\pi)\) is the representation of the Weil group of \(F\) associated to \(\pi\) by the Langlands correspondence, dualized and slightly twisted. Under certain assumptions, Carayol proved his conjecture for \(n=2\) and sketched a program to generalize his methods to higher dimensions. In this paper, the author carries out most of Carayol’s program for the part of the cohomology corresponding to supercuspidal representations of \(GL(n,F)\). The results of this paper include the realization of the generalized Jacquet-Langlands correspondence, and a partial description of the associated Weil group representations.

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
14F20 Étale and other Grothendieck topologies and (co)homologies

Citations:

Zbl 0704.11049
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