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

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.