Semisimple characters for inner forms II: Quaternionic inner forms of classical groups

Abstract: In this article we consider a quaternionic inner form $G$ of a $p$-adic classical group defined over a non-archimedian local field of odd residue characteristic. We construct all full self-dual semisimple characters for $G$ and we classify their intertwining classes using endo-parameters. Further we prove an intertwining and conjugacy theorem for self-dual semisimple characters. We give the formulas for the set of intertwiners between self-dual semisimple characters. We count all $G$-intertwining classes of self-dual semisimple characters which lift to the same $\tilde{G}$-intertwining class of a semisimple character for the ambient general linear group $\tilde{G}$ for $G$.