Depth zero representations over ℤ[(1)/(p)]

2022/02/08 by Dat, Jean-François, Lanard, Thomas
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2202.03982

Abstract

We consider the category of depth 0 representations of a p-adic quasi-split reductive group with coefficients in ℤ[(1)/(p)]. We prove that the blocks of this category are in natural bijection with the connected components of the space of tamely ramified Langlands parameters for G over ℤ[(1)/(p)]. As a particular case, this depth 0 category is thus indecomposable when the group is tamely ramified. Along the way we prove a similar result for finite reductive groups. As an application, we deduce that the semi-simple local Langlands correspondence π↦ φ_π constructed by Fargues and Scholze takes depth 0 representations to tamely ramified parameters, using a motivic version of their construction recently announced by Scholze. We also bound the restriction of φ_π to tame inertia in terms of the Deligne-Lusztig parameter of π and show, in particular, that φ_π is unramified if π is unipotent.

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