Cuspidal representations of \(\mathrm{GL}_r(D)\) distinguished by an inner involution (Q6596069)

Let \(K\) be a quadratic extension of a non-Archimedean compact field of characteristic residue \(p\ne 2\) the general linear group \(G=\mathrm{GL}_r(D)\) a division algebras \(D\) of reduced degree \(d\) such that \(rd = 2n\) with respect to a quadratic extension \(K\) of a non-Archimedean compact field of characteristic residue \(p\ne 2\). The author proves a conjecture by \textit{D. Prasad} and \textit{R. Takloo-Bighash} [J. Reine Angew. Math. 655, 189--243 (2011; Zbl 1228.11070)], namely Conjecture 1.1, about the value of a half of the epsilon factor of the restriction \(e_K(\phi)\) of the Langlands parameter \(\phi\) of the Weil-Deligne group of \(K\) to the centralizer \(H\) in \(G\). The main result is that \(e_K(\phi)= (-1)^r\) (Theorem 1.5, Proposition 1.6, Theorems 1.7--1.9)