Let n be a natural number, and let p and q be two non-negative integers with \(p+q = n\) . Let \({{\,\textrm{F}\,}}\) be a local non-Archimedean field of characteristic zero with a finite residue field. Set \(G_n = {{\,\textrm{GL}\,}}_n({{\,\textrm{F}\,}})\) . Consider the subgroup \(H_{p,q}\) of \(G_n\) defined as follows: \(\begin{aligned} H_{p,q} = \Bigg \{\begin{pmatrix} g_1 & 0 \\ 0 & g_2 \end{pmatrix} :~ g_1 \in G_p ~\text {and}~g_2 \in G_q\Bigg \}. \end{aligned}\) Then a complex smooth representation \((\pi ,V)\) of \(G_n\) is said to be \(H_{p,q}\) -distinguished (or said to have a linear period with respect to \(H_{p,q}\) ) if there exists a linear functional \(\psi \) on V such that \(\psi (\pi (h)v) = \psi (v)\) for all \(v \in V\) and \(h \in H_{p,q}\) . In this article, we classify those smooth irreducible representations of \(G_{4}\) that are \(H_{2,2}\) -distinguished. Furthermore, we provide a precise characterization of the symplectic Langlands parameters that correspond to \(H_{2,2}\) -distinguished representations of \(G_4\) .
