Branching laws for the Steinberg representation: the rank 1 case

TL;DR

This paper explores the branching laws of the Steinberg representation for rank 1 reductive symmetric spaces over p-adic fields, establishing a reciprocity law linking different intertwining space dimensions without explicitly computing them.

Contribution

It introduces a reciprocity law connecting the dimension of the Steinberg representation's intertwining space with those of certain anisotropic subgroups, providing a new perspective on branching problems.

Findings

Relates the dimension of ${ m Hom}_H ( ext{St}_G, ext{π})$ to other intertwining spaces.

Establishes a reciprocity law for branching problems in rank 1 symmetric spaces.

Does not compute the dimensions explicitly, but links them through a new theoretical framework.

Abstract

Let $G/H$ be a reductive symmetric space over a $p$-adic field $F$, the algebraic groups $G$ and $H$ being assumed semisimple of relative rank $1$. One of the branching problems for the Steinberg representation $\St_G$ of $G$ is the determination of the dimension of the intertwining space ${\rm Hom}_H (\St_G ,\pi )$, for any irreducible representation $\pi$ of $H$. In this work we do not compute this dimension, but show how it is related to the dimensions of some other intertwining spaces ${\rm Hom}_{K_i} ({\tilde \pi} ,1)$, for a certain finite family $K_i$, $i=1,...,r$, of anisotropic subgroups of $H$ (here ${\tilde \pi}$ denote the contragredient representation, and $1$ the trivial character). In other words we show that there is a sort of `reciprocity law' relating two different branching problems.