Projective smooth representations in natural characteristic

2024/11/19 by Ophir, Amit, Sorensen, Claus
#11F70 #20C20 #22E50 #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2411.12867

Abstract

We investigate under which circumstances there exists nonzero \itprojective smooth \field[G]-modules, where \field is a field of characteristic p and G is a locally pro-p group. We prove the non-existence of (non-trivial) projective objects for so-called \itfair groups -- a family including \bfG(\frakF) for a connected reductive group \bfG defined over a non-archimedean local field \frakF. This was proved in \citeSS24 for finite extensions \frakF/ℚ_p. The argument we present in this note has the benefit of being completely elementary and, perhaps more importantly, adaptable to \frakF=\BbbF_q( (t) ). Finally, we elucidate the fairness condition via a criterion in the Chabauty space of G.

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