Abstract
We study admissible orthogonal and symplectic representations of the Weil-Deligne group W′( $$ \overline K $$ /K) of a local non-Archimedean field K. As an application of the obtained results we show that the root number of the tensor product of two admissible symplectic representations of W′( $$ \overline K $$ /K) is 1.
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