Symmetry breaking for $\operatorname{PGL}(2)$ over non-archimedean local fields

TL;DR

This paper constructs explicit symmetry breaking operators between principal series representations of PGL(2) over quadratic extensions of non-archimedean local fields, classifies all such operators, and explores their properties and implications.

Contribution

It provides explicit distribution kernels for symmetry breaking operators and classifies all intertwining operators between principal series representations of PGL(2) over local fields.

Findings

Explicit holomorphic families of intertwining operators constructed.

Support and mapping properties of distributions determined.

Every Steinberg representation of PGL(2,E) contains a PGL(2,F) Steinberg as a summand.

Abstract

For a quadratic extension $\mathbb{E}/\mathbb{F}$ of non-archimedean local fields we construct explicit holomorphic families of intertwining operators between principal series representations of $\operatorname{PGL}(2,\mathbb{E})$ and $\operatorname{PGL}(2,\mathbb{F})$, also referred to as symmetry breaking operators. These families are given in terms of their distribution kernels which can be viewed as distributions on $\mathbb{E}$ depending holomorphically on the principal series parameters. For all such parameters we determine the support of these distributions, and we study their mapping properties. This leads to a classification of all intertwining operators between principal series representations, not necessarily irreducible. As an application, we show that every Steinberg representation of $\operatorname{PGL}(2,\mathbb{E})$ contains a Steinberg representation of $\operatorname{PGL}(2,\mathbb{F})$ as a direct summand of Hilbert spaces.