Branching laws for Stein's complementary series and Speh representations of $\operatorname{GL}(2n,\mathbb{R})$

TL;DR

This paper explicitly decomposes Stein's complementary series and Speh representations of GL(2n,R) when restricted to GL(2n-1,R), revealing their structure as integrals of induced representations from specific parabolic subgroups.

Contribution

It provides the first explicit direct integral decomposition of these representations upon restriction, using symmetry breaking operators and meromorphic analysis.

Findings

Decomposition expressed as direct integrals of induced representations.

Identification of symmetry breaking operators with detailed meromorphic properties.

Alignment with the theory of adduced representations.

Abstract

We obtain the explicit direct integral decomposition of Stein's complementary series representations and Speh representations of $\operatorname{GL}(2n,\mathbb{R})$ when restricted to the subgroup $\operatorname{GL}(2n-1, \mathbb{R})$. The decomposition is a direct integral of unitarily induced representations from a maximal parabolic subgroup of $\operatorname{GL}(2n-1, \mathbb{R})$ with Levi factor $\operatorname{GL}(2n-2, \mathbb{R})\times\operatorname{GL}(1, \mathbb{R})$, where the induction data consists of a complementary series or Speh representation of the factor $\operatorname{GL}(2n-2, \mathbb{R})$ with the same parameter as the one of $\operatorname{GL}(2n, \mathbb{R})$ and a character of $\operatorname{GL}(1, \mathbb{R})$. These results are in line with the theory of adduced representations. The main tools in the proof are two families of symmetry breaking operators between degenerate series representations of $\operatorname{GL}(2n, \mathbb{R})$ and $\operatorname{GL}(2n-1, \mathbb{R})$ whose meromorphic properties are studied in great detail.