In this paper, we prove that the depths of irreducible admissible representations are preserved by the local theta correspondence for any type I reductive dual pairs over a nonarchimedean local field.
"Depth preservation in local theta correspondence." Duke Math. J. 113 (3) 531 - 592, 15 June 2002. https://doi.org/10.1215/S0012-7094-02-11334-9