On representations of the Weil–Deligne group

We study admissible orthogonal and symplectic representations of the Weil–Deligne group $\mathcal{W}'(\overline K/K)$ 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 $\mathcal{W}'(\overline K/K)$ is 1.