On the Moy-Prasad Filtration and Stable Vectors

Abstract

Let K be a maximal unramified extension of a nonarchimedean local field of residual characteristic p > 0. Let G be a reductive group over K which splits over a tamely ramified extension of K. To a point x in the Bruhat–Tits building of G over K, Moy and Prasad have attached a filtration of G(K) by bounded subgroups.

In this thesis, we give necessary and sufficient conditions for the existence of stable vectors in the dual of the first Moy–Prasad filtration quotient V_x under the action of the reductive quotient G_x. This extends earlier results by Reeder and Yu for large residue-field characteristic and yields new supercuspidal representations for small primes p.

Moreover, we show that the Moy–Prasad filtration quotients for different residue-field characteristics agree as representations of the reductive quotient in the following sense: For some N coprime to p, there exists a representation of a reductive group scheme over Spec(Z[1/N]) all of whose special fibers are Moy–Prasad filtration representations. In particular, the special fiber above p corresponds to G_x acting on V_x.

In addition, we provide a new description of the representation of G_x on V_x as a representation occurring in a generalized Vinberg–Levy theory. This generalizes an earlier result by Reeder and Yu for large primes p. Moreover, we describe these representations in terms of Weyl modules.

In this thesis, we also treat reductive groups G that are more general than those that split over a tamely ramified field extension of K.