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

We obtain the explicit direct integral decomposition of Stein’s complementary series representations and Speh representations of $\mathrm{GL}(2n,\mathbb{R})$ when restricted to the subgroup $\mathrm{GL}(2n-1,\mathbb{R})$. The decomposition is a direct integral of unitarily induced representations from a maximal parabolic subgroup of $\mathrm{GL}(2n-1,\mathbb{R})$ with Levi factor $\mathrm{GL}(2n-2,\mathbb{R})\times \mathrm{GL}(1,\mathbb{R})$, where the induction data consists of a complementary series or Speh representation of the factor $\mathrm{GL}(2n-2,\mathbb{R})$ with the same parameter as the one of $\mathrm{GL}(2n,\mathbb{R})$ and a character of $\mathrm{GL}(1,\mathbb{R})$. 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 $\mathrm{GL}(2n,\mathbb{R})$ and $\mathrm{GL}(2n-1,\mathbb{R})$ whose meromorphic properties are studied in great detail.