The Stone-von Neumann Construction in Branching Rules and Minimal Degree Problems

Abstract

In Part I, we investigate the principal series representations of the n-fold covering groups of the special linear group over a p-adic field. Such representations are constructed via the Stone-von Neumann theorem. We have three interrelated results. We first compute the K-types of these representations. We then give a complete set of reducibility points for the unramified principal series representations. Among these are the unitary unramified principal series representations, for which we further investigate the distribution of the K-types among its irreducible components. In Part II, we demonstrate another application of the Stone-von Neumann theorem. Namely, we present a lower bound for the minimal degree of a faithful representation of an adjoint Chevalley group over a quotient ring of a non-Archimedean local field.