Explicit Structure and Data for Supercuspidal Representations of SO_5

Abstract

This thesis presents an explicit classification of the structural data needed for the construction of supercuspidal representations for the split special orthogonal group $\Sos$ over a non-archimedean local field $F$. While general constructions of supercuspidal representations, such as that of Yu, are well established, their application to specific groups requires detailed knowledge of objects that are often not available in an explicit form. We classify the elliptic maximal tori of $\Sos$ and describe their embeddings into twisted Levi subgroups, which are themselves classified via both root-theoretic and Lie algebraic methods. We determine the non-regular elements in the Lie algebras of these tori and compute their centralizers, providing a second, independent verification of the twisted Levi subgroups. We further determine the Moy--Prasad filtrations of each toral Lie algebra and study the genericity conditions essential to Yu's construction. We establish precise criteria for so-called generic elements in terms of valuations and eigenvalue data, and determine the admissible sequences in the $G$-datum. We then develop explicit methods to determine the corresponding points in the Bruhat--Tits building by constructing explicit embeddings of each elliptic torus, which also allows us to compute the reductive quotients associated to relevant twisted Levi subgroup. We present the results in a series of tables and diagrams that provide the necessary inputs for the construction of supercuspidal representations for ease of reference for practioners in the field.We also include a comparison with the corresponding data for $\mathrm{Sp}_4$.