Let \(D\) be a central division algebra over a number field \(k\) and consider the reductive \(k\)-group \(G = \mathrm{GL}_{2m}(D)\) with its subgroup \(H = \mathrm{GL}_m(D)^2\). The periods of an element in a cuspidal irreducible automorphic representation \(\pi\) of \(G(\mathbb{A}_k)\) with respect to \(H(\mathbb{A}_k)\) are called “linear periods” in this paper. Automorphic representations \(\pi\) with a nonzero \(H(\mathbb{A}_k)\)-period are called \(H\)-distinguished. Let \(G' = \mathrm{GL}_{2n}\) and \(H' = \mathrm{GL}_n^2\) be the split inner forms of \(G\) and \(H\), respectively, to which one can attach the Jacquet-Langlands transfer \(\pi'\) on \(G'(\mathbb{A}_k)\); similarly, there is a notion of \(H'\)-distinction for \(\pi'\). The following conjecture is proposed in this paper: \(\pi\) is \(H\)-distinguished if and only if \(\pi'\) is \(H'\)-distinguished.
The paper proposes an approach towards this conjecture based on comparison of relative trace formulas. A key ingredient thereof is established: the smooth transfer of test functions in the non-Archimedean local set-up. The proof is divided into two steps: (i) linearize the problem by passing to Lie algebras; (ii) prove that the smooth transfer commutes with Fourier transforms up to an explicit constant. The author combines some techniques in [J. L. Waldspurger, Compos. Math. 105, No. 2, 153–236 (1997; Zbl 0871.22005)] together with those in [W. Zhang, Ann. Math. (2) 180, No. 3, 971–1049 (2014; Zbl 1322.11048)] on Jacquet-Rallis trace formula. Some results on harmonic analysis of the symmetric spaces \(H \backslash G\), \(H' \backslash G'\) are also used. Note that one does not need a relative “fundamental lemma” in this case.