Orbital Integrals on Unitary Hyperbolic Spaces Over $\frak p$-adic Fields

Abstract

For a given étale quadratic algebra $E$ over a $\mathfrak{p}$-adic field $F$, we establish a transfer of unramified test functions on the symmetric space $\mathrm{GL}(2,F)\backslash\mathrm{GL}(2,E)$ to those on a unitary hyperbolic space so that the orbital integrals match. This is an important step toward a comparison of relative trace formulas of these symmetric spaces, which would yield an example of a non-tempered analogue of a refined global Gross-Prasad conjecture.