Uniqueness of Rankin-Selberg periods

Let $k$ be a local field of characteristic zero. Rankin-Selbergs local zeta integrals produce linear functionals on generic irreducible admissible smooth representations of $GL_n(k)times GL_r(k)$, with certain invariance properties. We show that up to scalar multiplication, these linear functionals are determined by the invariance properties.

Multiplicity preservation for orthogonal-symplectic and unitary dual pair correspondences

Over a non-archimedean local field of characteristic zero, we prove the multiplicity preservation for orthogonal-symplectic dual pair correspondences and unitary dual pair correspondences.