Parity of the Langlands parameters of conjugate self-dual representations of GL(n) and the local Jacquet-Langlands correspondence

We determine the parity of the Langlands parameter of a conjugate self-dual supercuspidal representation of GL(n) over a non-archimedean local field by means of the local Jacquet-Langlands correspondence. It gives a partial generalization of a previous result on the self-dual case by Prasad and Ramakrishnan.