#### Vol. 13, No. 7, 2019

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A vanishing result for higher smooth duals

### Claus Sorensen

Vol. 13 (2019), No. 7, 1735–1763
DOI: 10.2140/ant.2019.13.1735
##### Abstract

In this paper we prove a general vanishing result for Kohlhaase’s higher smooth duality functors ${S}^{i}$. If $G$ is any unramified connected reductive $p$-adic group, $K$ is a hyperspecial subgroup, and $V$ is a Serre weight, we show that ${S}^{i}\left({ind}_{K}^{G}V\right)=0$ for $i>dim\left(G∕B\right)$, where $B$ is a Borel subgroup and the dimension is over ${ℚ}_{p}$. This is due to Kohlhaase for ${GL}_{2}\left({ℚ}_{p}\right)$, in which case it has applications to the calculation of ${S}^{i}$ for supersingular representations. Our proof avoids explicit matrix computations by making use of Lazard theory, and we deduce our result from an analogous statement for graded algebras via a spectral sequence argument. The graded case essentially follows from Koszul duality between symmetric and exterior algebras.

##### Keywords
mod p representations, higher smooth duality, Lazard theory
Primary: 20C08
Secondary: 22E50