Local theta correspondence of depth zero representations and theta dichotomy

In this paper, we prove that depth zero representations are preserved by local theta correspondence for any type I reductive dual pairs over a p-adic field. Moreover, the minimal K-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 p-adic reductive dual pairs of symplectic and orthogonal group, which is analogous to S. Kudla and S. Rallis' result for p-adic unitary groups.