A vanishing result for higher smooth duals

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(G∕B), where B is a Borel subgroup and the dimension is over ℚp. This is due to Kohlhaase for GL2(ℚp), 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.