Item  and  explore the relationship between derived functor modules and local theta correspondence over real numbers. In , we investigate the transfer of the theta lift of a unitary lowest weight module of a symplectic group and show the transfer is in fact also a theta lift of a certain unitary lowest weight module. In , I extend the theorem to all possible dual pairs and simplify the proof by using an identity on Hecke algebras. In addition to these, we have shown that the lifts in  are quotients of certain .
Item [i] and  study the associated cycle of the theta lift. In [i], we exam the lift of unitary lowest weight module without restricting on the stable range. When beyond stable range, our results exhibit the subtlety of the problem. Along the discussion some technical lemmas are established which serve as preparations for . In , the associated cycle of a full (big) theta lift in the stable range is completely described in terms of the associated cycle of the original representation. Moreover, we show that the big theta lift of a unitary representation is in fact irreducible in the stable range. As a corollary, we obtain a class of unipotent representations via iterated theta lifting which satisfy a conjecture of Vogan on -types.
Item [ii] is a summary of results in  and [i] of the case of theta lifts of one dimensional representations.
Item  and  study theta correspondence between supercuspidal representations. In , we consider epipelagic supercuspidal representations and show that our construction exhaust all possible cases under a mild condition of the residual characteristic. We also explored some facts about the Bruhat-Tits building of classical groups, lattice functions and their compatibility with moment maps . In , we define a notion of theta lift of (tamely ramified) supercuspidal data and show that local theta correspondence between supercuspidal representations are completely described by the lift of data under the assumption that the residual characteristic is large enough. In appendix B of , we gives a short proof of “depth preservation”.
Item , we investigate the attempts of using certain symmetry condition/tautological functorial lift to characterize the smooth irreducible representations of GL(n,E) that are distinguished by its subgroup GL(n,F), where E/F is a quadratic extension of p-adic fields. We show that the attempt works for ladder representations. On the other hand, we show that such kind of approach does not work for general admissible representations, by constructing a counter example using irreducibly induced representation from ladders.