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 Si. If G is any unramified connected reductive p-adic group, K is a hyperspecial subgroup, and V is a Serre weight, we show that Si(indKGV ) = 0 for i > dim(GB), where B is a Borel subgroup and the dimension is over p. This is due to Kohlhaase for GL2(p), in which case it has applications to the calculation of Si 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
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