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In this paper, we completely describe the Howe correspondence for the dualpairs from the title over a nonarchimedean local field of characteristic zero.More specifically, for every irreducible admissible representation of thesegroups, we find its first occurrence index in the theta correspondence and wedescribe, in terms of their Langlands parameters, the small theta lifts on alllevels.

Theta correspondence for symplectic-orthogonal and metaplectic-orthogonal p-adic dual pairs