In this paper, we prove that depth zero representations are preserved by local theta correspondence for any type I reductive dual pairs over a -adic field. Moreover, the minimal -types of the paired depth zero irreducible admissible representations are paired by the theta correspondence for finite reductive dual pairs. As a consequence, we prove that the Iwahori-spherical representations are preserved by the local theta correspondence. Then we obtain some partial result of theta dichotomy for finite reductive dual pairs and -adic reductive dual pairs of symplectic and orthogonal group, which is analogous to S. Kudla and S. Rallis' result for -adic unitary groups.
Shu-Yen PAN. "Local theta correspondence of depth zero representations and theta dichotomy." J. Math. Soc. Japan 54 (4) 793 - 845, October, 2002. https://doi.org/10.2969/jmsj/1191591993