On non-basic Rapoport-Zink spaces

Mantovan, Elena

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 5, pp. 671-716.

Dans cet article, on considère certains espaces de Rapoport-Zink non-ramifiés, associés à des groupes $p$-divisibles non-basiques et on étudie leur géométrie vis-à-vis de celle des espaces de Rapoport-Zink basiques correspondants. L’origine de ce problème se situe, d’une part, dans la conjecture de Kottwitz concernant la réalisation des correspondances de Langlands locales dans la cohomologie étale $l$-adique des espaces de Rapoport-Zink et, d’autre part, plus simplement dans la question d’identifier pour lesquels de ces espaces la partie supercuspidale de la cohomologie n’est pas vide. Nos résultats sont compatibles avec cette conjecture et, dans certains cas particuliers, ils répondent à la dernière question. En particulier, dans ces cas, on établit une formule reliant la cohomologie de ces espaces à l’induction parabolique de celle de certains espaces de Rapoport-Zink de dimension inférieure (et dans les cas plus favorables basiques). Cette formule a été précédemment conjecturée par Harris dans [11] (Conjecture 5.2, p. 420).

In this paper we study certain moduli spaces of Barsotti-Tate groups constructed by Rapoport and Zink as local analogues of Shimura varieties. More precisely, given an isogeny class of Barsotti-Tate groups with unramified additional structures, we investigate how the associated (non-basic) moduli spaces compare to the (basic) moduli spaces associated with its isoclinic constituents. This aspect of the geometry of the Rapoport-Zink spaces is closely related to Kottwitz’s prediction that their $l$-adic cohomology groups provide a realization of certain cases of local Langlands correspondences and in particular to the question of whether they contain any supercuspidal representations. Our results are compatible with this prediction and identify many cases when no supercuspidal representations appear. In those cases, we prove that the $l$-adic cohomology of the non-basic spaces is equal (in the appropriate sense) to the parabolic induction of the $l$-adic cohomology of some associated lower-dimensional (and in the most favorable cases basic) Rapoport-Zink spaces. Such an equality was originally conjectured by Harris in [11] (Conjecture 5.2, p. 420).

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