Distinction inside L-packets of SL(n)

If $E/F$ is a quadratic extension $p$-adic fields, we first prove that the $\mathrm{SL}_n(F)$-distinguished representations inside distinguished unitary L-packet of $\mathrm{SL}_n(E)$ are precisely those admitting degenerate Whittaker model with respect to character $N(E)/N(F)$. Then establish global analogue this result. For this, let be number fields and $\pi$ an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished square integrable automorphic representation $\mathrm{SL}_n(\mathbb{A}_E)$. Let $(\sigma,d)$ unique pair associated $\pi$, where $\sigma$ cuspidal $\mathrm{GL}_r(\mathbb{A}_E)$ $n=dr$. Using unfolding argument, element $\mathrm{SL}_n(\mathbb{A}_F)$ if only it has for $\psi$ type $r^d:=(r,\dots,r)$ $N_n(\mathbb{A}_E)$ which trivial on $N_n(E+\mathbb{A}_F)$, $N_n$ group unipotent upper triangular matrices $\mathrm{SL}_n$. As application, under assumptions splits at infinity $r$ odd, local-global principle $\mathrm{SL}_n(\mathbb{A}_F)$-distinction $\pi$. second application construct examples $\mathrm{SL}_n(\mathbb{A}_E)$ such period integral vanishes some canonical copy everywhere locally their L-packets do not contain any representation.