| Popis: |
Let F be a locally compact nonarchimedean field with residue characteristic p and G the group of F -rational points of a connected split reductive group over F . We define a torsion pair in the category Mod ( H ) of modules over the pro- p -Iwahori Hecke k -algebra H of G , where k is an arbitrary field. We prove that, under a certain hypothesis, the torsionfree class embeds fully faithfully into the category Mod I ( G ) of smooth k -representations of G generated by their pro- p -Iwahori fixed vectors. If the characteristic of k is different from p then this hypothesis is always satisfied and the torsionfree class is the whole category Mod ( H ) . If k contains the residue field of F then we study the case G = S L 2 ( F ) . We show that our hypothesis is satisfied, and we describe explicitly the torsionfree and the torsion classes. If F ≠ Q p and p ≠ 2 , then an H -module is in the torsion class if and only if it is a union of supersingular finite length submodules; it lies in the torsionfree class if and only if it does not contain any nonzero supersingular finite length module. If F = Q p , the torsionfree class is the whole category Mod ( H ) , and we give a new proof of the fact that Mod ( H ) is equivalent to Mod I ( G ) . These results are based on the computation of the H -module structure of certain natural cohomology spaces for the pro- p -Iwahori subgroup I of G . |
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