## Publications and preprints

- Transfers of K-types on local theta lifts of characters and unitary lowest weight modules,

(with Hung Yean Loke and U-Liang Tang), ArXiv e-prints:1207.6454,

Israel Journal of Mathematics, Vol 201, Issue 1, pp 1-24, 20 June 2014 - Derived functor modules, dual pairs and U(𝔤)
^{K}-actions, ArXiv e-prints:1310.6378, Journal of Algebra, Volume 450, pp 629–645, Mar 2016 - Invariants and K-spectrums of local theta lifts,

(with Hung Yean Loke), ArXiv e-prints:1302.1031, Compositio Mathematica, Jan 2015 - Local theta correspondences between epipelagic supercuspidal representations,

(with Hung Yean Loke and Gordan Savin), ArXiv e-prints:1501.07069, Mathematische Zeitschrift, Jun 2016 - Local theta correspondence between supercuspidal representations,

(with Hung Yean Loke), ArXiv e-prints: 1512.01797, 2017, accepted by Annales Scientifiques de l’ENS - On two questions concerning representations distinguished by the Galois involution,

(with Maxim Gurevich and Arnab Mitra), ArXiv e-prints: 1609.03155, 2017, Forum Mathematicum - Special unipotent representations : orthogonal and symplectic groups (with Dan Barabasch, Binyong Sun and Chen-Bo Zhu) ArXiv e-prints: 1712.05552

(new version). The python code of the algorithm in the paper can be found at

https://github.com/jiajunma/unipotentrepn - Nilpotent orbits of orthogonal groups over p-adic fields, and the DeBacker parametrization (with Tobias Bernstein, Monica Nevins, Jit Wu Yap) ArXiv e-prints:1808.03593

## Notes

- Associated cycles of local theta lifts of unitary characters and unitary lowest weight modules,

(with Hung Yean Loke and U-Liang Tang), ArXiv e-prints:1207.6451, 2011 - Local Theta lifts of one-dimensional representations, Notes on Symposium on Representation Theory 2012, Kagoshima, Japan

## Descriptions

Item [1] and [2] explore the relationship between derived functor modules and local theta correspondence over real numbers. In [1], 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 [2], 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 [1] are quotients of certain .

Item [i] and [3] 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 [3]. In [3], 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 [1] and [i] of the case of theta lifts of one dimensional representations.

Item [4] and [5] study theta correspondence between supercuspidal representations. In [4], 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 [4]. In [5], 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 [5], we gives a short proof of “depth preservation”.

Item [6], 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.