On the Construction of Supercuspidal Representations

Abstract

This thesis contributes to the construction of supercuspidal representations in small residual characteristics. Let G be a connected, quasi-split, semisimple reductive algebraic group defined and quasi-split over a non-archimedean local field k and splitting over a tamely, totally ramified extension of k. To each parahoric subgroup of G(k), Moy and Prasad have attached a natural filtration by compact open subgroups, the first of which is called the pro-unipotent radical of the parahoric subgroup. The first main result of this thesis is to characterize shallow characters of a pro-unipotent radical, those being complex characters that vanish on the smallest Moy-Prasad subgroup containing all commutators of linearly-dependent affine k-root groups. Through low-rank examples, we illustrate how this characterization can be used to explicitly construct all shallow characters. Next, we provide a natural sufficient condition under which a shallow character compactly induces as a direct sum of supercuspidal representations of G(k). Through examples, however, we show that this sufficient condition need not be necessary, all while constructing new supercuspidal representations of Sp_4(k) when p = 2 and the split form of G_2 over k when p = 3. This work extends the construction of the simple supercuspidal representations given by Gross and Reeder and the epipelagic supercuspidal representations given by Reeder and Yu.