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Orbital integrals on unitary hyperbolic spaces over $$\mathfrak p$$-adic fields. (English) Zbl 1404.22023
Summary: 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 $$\mathrm{GL}(2,F)\mathrm{GL}(2,E)$$ 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.

##### MSC:
 22E35 Analysis on $$p$$-adic Lie groups 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E50 Representations of Lie and linear algebraic groups over local fields
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