Multiplicity one for a linear spherical pair and its unitary analogue

Abstract

We prove multiplicity one for a linear spherical pair associated with $\mathrm{GL}_4(F)\times\mathrm{GL}_2(F)$ and its unitary similitude analogue associated with $\mathrm{GU}(2,2)(F)\times\mathrm{GU}(1,1)(F)$. Both results hold for all irreducible admissible representations over non-Archimedean local fields of characteristic zero. Generalized Harish-Chandra descent reduces the proofs to the analysis of closed orbits and their normal representations. The unitary proof combines reductions to the linear case with a rational classification of closed orbits and a separate analysis of the remaining normal representations. We also establish multiplicity one for unitary Shalika periods of $\mathrm{GU}(2,2)(F)$ with arbitrary irreducible admissible coefficients. A theta-period identity relates these periods to the unitary spherical pair. By varying the splitting character, we prove that every irreducible admissible representation of $\mathrm{GU}(2,2)(F)$ occurs as a quotient of a suitable big theta lift, allowing the multiplicity-one bound to pass to the Shalika periods. The linear argument also recovers Rallis's multiplicity-one theorem for twisted Jacquet modules of $\mathrm{GL}_4(F)$.