PARITY OF CONJUGATE SELF-DUAL REPRESENTATIONS OF INNER FORMS OF $\operatorname { {GL}}_n$ OVER <i>p</i>-ADIC FIELDS

Abstract We prove a general formula that relates the parity of the Langlands parameter of a conjugate self-dual discrete series representation of $\operatorname { {GL}}_n$ to the parity of its Jacquet-Langlands image. It gives a generalization of a partial result by Mieda concerning the case of invariant $1/n$ and supercuspidal representations. It also gives a variation of the result on the self-dual case by Prasad and Ramakrishnan.