There are two parametrizations of discrete series representations of GL_n over p-adic fields. One is the local Langlands correspondence, and the other is the local Jacquet-Langlands correspondence. The composite of these two maps the discrete series representations of an inner form of GL_n to Galois representations called discrete L-parameters. On the other hand, we can define the parity for each self-dual representation depending on whether the representation is orthogonal or symplectic. The composite preserves the notion of self-duality, and it transforms the parity in a nontrivial manner. Prasad and Ramakrishnan computed the transformation law, and Mieda proved its conjugate self-dual analog under some conditions on groups and representations. We will talk about the proof of the general case of this analog. We use the globalization method, as in the proof of Prasad and Ramakrishnan.

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