Correspondence of discrete series representations of ${\mathrm{G}\mathrm{S}\mathrm{p}}_{2n}$ and its inner form

We construct a correspondence with a character equation of discrete series representations of ${\mathrm{G}\mathrm{S}\mathrm{p}}_{2n}$ over a $p$-adic field and its inner form for $p>2$ and $n\ge 2$. This is a generalization of the works by Chan and Gan and the local Jacquet-Langlands correspondence. Moreover, our results partially reveal what the local Langlands correspondence of ${\mathrm{G}\mathrm{S}\mathrm{p}}_{2n}$ and its inner form over $p$-adic fields should be.