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

Abstract

We construct a correspondence with a character equation of discrete series representations of $\mathrm{GSp}_{2n}$ over a $p$-adic field and its inner form for $p>2$ and $n\geq 2$. 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 $\mathrm{GSp}_{2n}$ and its inner form over $p$-adic fields should be.