Branching Laws for Stein’s Complementary Series and Speh Representations of $$\textrm{GL}(2n,\mathbb {R})$$

Abstract

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