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We construct and study the holomorphic discrete series representation and theprincipal series representation of the symplectic group $\mathrm{Sp}(2n,F)$over a $p$-adic field $F$ as well as a duality between some sub-representationsof these two representations. The constructions of these two representationsgeneralize those defined in Morita and Murase's works. Moreover, Morita built aduality for $\mathrm{SL}(2, F)$ defined by residues. We view the duality wedefined as an algebraic interpretation of Morita's duality in some extent andits generalization to the symplectic groups.

Morita's Theory for the Symplectic Groups