Branching Rules for Irreducible Smooth Representations of Unramified U(1,1)

Abstract

Let G = U(1,1) denote the group of F-points of the quasi-split E/F-form of GL₂, where F is a non-archimedean local field of residual characteristic p ≠ 2, and E is the quadratic unramified extension of F. In this thesis, we determine the branching of almost all irreducible smooth representations of G upon restriction to a fixed maximal compact subgroup K. We prove that each such restriction decomposes as a multiplicity-free direct sum of irreducible components of distinct depth and degree, up to twisting by a quasi-character of G. Moreover, we give an explicit description of all irreducible components that occur in this decomposition in terms of irreducible representations of K constructed herein.

We analyze the branching rules by dividing the irreducible representations of G into three classes: depth-zero supercuspidal representations, positive-depth supercuspidal representations, and principal series representations. We provide two applications of this explicit description. First, we show that the higher-depth components arising in these decompositions exhibit a striking uniformity: up to twisting by a quasi-character of G, they coincide with the higher-depth components obtained from a fixed collection of four depth-zero irreducible supercuspidal representations. Second, we prove that the restriction of irreducible representations of G to a smaller subgroup of K can be described entirely in terms of the trivial representation and certain representations arising from nilpotent orbits in the Lie algebra of G, thereby establishing a new case of a recent conjecture in the literature.