A vanishing result for higher smooth duals

Sorensen, Claus

doi:10.2140/ant.2019.13.1735

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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 $S^{i} ({ind}_{K}^{G} V) = 0$ for $i > dim (G ∕ B)$ , where $B$ is a Borel subgroup and the dimension is over $ℚ_{p}$ . This is due to Kohlhaase for ${GL}_{2} (ℚ_{p})$ , 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.

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Keywords

mod p representations, higher smooth duality, Lazard theory

Mathematical Subject Classification 2010

Primary: 20C08

Secondary: 22E50

Milestones

Received: 1 October 2018

Revised: 27 March 2019

Accepted: 21 May 2019

Published: 21 September 2019

Authors

Claus Sorensen
Department of Mathematics University of California, San Diego La Jolla, CA United States

Vol. 13, No. 7, 2019