Email: ting.xue at unimelb(dot)edu(dot)au
Office: Peter Hall building 203
Phone: +61 (0)3 8344 2182
Research interest – Representation theory, algebraic groups, geometry of nilpotent orbits, Springer theory, combinatorics arising from representation theory.
The (generalised) Springer theory for reductive groups plays an important role in representation theory. It relates nilpotent orbits to irreducible representations of Weyl groups. In previous work [3,4,5,7,13] we describe the Springer correspondence for (classical) Lie algebras and their duals in bad characteristics. Many questions in algebraic groups have rather uniform answers over fields of large characteristic. When the characteristic of the base field is small, special interesting phenomena can occur. In [8,9,11] we study the nilpotent adjoint and coadjoint orbits in small characteristic and their properties. In particular we describe the nilpotent pieces (following Lusztig). These pieces have nice properties independent of the characteristic of the base field.
My current research focus is to establish a Springer theory in the setting of graded Lie algebras together with my collaborators. In [15,19] we develop such a theory for symmetric pairs and classify the character sheaves (Fourier transforms of simple equivariant perverse sheaves supported on the nilpotent cone) arising in this setting. This in turn relies on a key geometric construction  of nearby cycle sheaves. As one application, we obtain a strategy to compute cohomology of Hessenberg varieties. The strategy was carried out in  for a special class of Hessenberg varieties, certain moduli spaces of vector bundles with extra structure on a hyper-elliptic curve. In the context of graded Lie algebras, Hecke algebras at roots of unity associated to complex reflection groups enter the description of character sheaves. In  we describe (conjecturally all) full support character sheaves explicitly in the case of stably graded classical Lie algebras. We make use of the nearby cycle sheaf construction for stable polar representations established in .
Most of my papers are available on arXiv.
25. (With K. Vilonen) Invariant systems and character sheaves for graded Lie algebras arXiv:2111.08403.
24. Character sheaves for symmetric pairs: spin groups. arXiv:2111.00403
23. (With K. Vilonen) Character sheaves for symmetric pairs: special linear groups. arXiv:2110.13451
22. (With K. Vilonen) A note on Hessenberg varieties. arxiv:2101.08652.
21. (With M. Grinberg and K. Vilonen) Nearby cycle sheaves for stable polar representations. arXiv:2012.14522.
20. (With K. Vilonen) Character sheaves for graded Lie algebras: stable gradings. arXiv:2012.08111.
18. (With M. Grinberg and K. Vilonen) Nearby cycle sheaves for symmetric pairs. arXiv:1805.02794.
17. (With T.H. Chen and K.Vilonen) Springer correspondence, Hyperelliptic curves, and cohomology of Fano varieties. Math. Res. Let. 27 (2020), no. 5, 1281–1323.
16. (With T.H. Chen and K. Vilonen) Hessenberg varieties, intersections of quadrics, and the Springer correspondence. Trans. AMS. 373 (2020), no.4, 2427–2461.
15. (With T.H. Chen and K. Vilonen) Springer correspondence for the split symmetric pair in type A. Compos. Math.154 (2018), no.11, 2403–2425.
14. (With T.H. Chen and K. Vilonen) On the cohomology of Fano varieties and the Springer correspondence. With an appendix by Dennis Stanton. Adv. in Math 318 (2017) 515-533.
13. Springer correspondence for exceptional Lie algebras and their duals in small characteristic. J. Lie Theory 27 (2017), no. 2, 357–375.
12. (With K. Vilonen) The null-cone and cohomology of vector bundles on flag manifolds. Represent. Theory 20 (2016), 482–498 (electronic).
11. Nilpotent coadjoint orbits in small characteristic. J. Algebra 397 (2014) 111–140.
10. (With G. Lusztig) Elliptic Weyl group elements and unipotent isometries with p=2. Represent. Theory 16 (2012), 270–275 (electronic).
9. Nilpotent pieces in the dual of odd orthogonal Lie algebras. Transform. Groups (2012) no. 2, 571–592.
8. On unipotent and nilpotent pieces for classical groups. J. Algebra 349 (2012), 165–184.
7. Combinatorics of the Springer correspondence for classical Lie algebras and their duals in characteristic 2. Adv. in Math 230 (2012), 229–262.
6. (With G. Lusztig) Appendix A: Unipotent elements in small characteristic III. J. Algebra 329 (2011) 187–189.
5. Nilpotent orbits in the dual of classical Lie algebras in characteristic 2 and the Springer correspondence. Represent. Theory 13 (2009), 609–635 (electronic).
4. Nilpotent orbits in classical Lie algebras over finite fields of characteristic 2 and the Springer correspondence. Represent. Theory 13 (2009), 371–390 (electronic).
3. Nilpotent orbits in classical Lie algebras over and the Springer correspondence. Proc. Natl. Acad. Sci. USA 105 (2008), no. 4, 1126–1128.
2. (With Y. Zhang) Bihamiltonian systems of hydrodynamic type and reciprocal transformations. Lett. Math. Phys. 75 (2006), no. 1, 79–92.
1. On a reduction of the Kadomtsev-Petviashvili hierarchy. Tsinghua Sci. Technol. 11 (2006), no. 1, 111–116.
Semester 1, Fall 2021 MAST30005 Algebra MAST10005 Calculus I
Semester 1, Fall 2020 MAST30005 Algebra MAST10005 Calculus I
Semester 2, Spring 2019 MAST90017 Representation theory
Semester 2, Spring 2016 MAST90097 Algebraic geometry
- Representation theory’s hidden motives, Sep 27-Oct 1, 2021
- MIT Lie groups seminar, March 17, 2021.
- Sydney Mathematical Research Institute Domestic Visitor, 2021. (Postponed)
- AMSI Winter School 2020, New directions in representation theory, University of Queensland. (Postponed.)
- Springer Fibres and Geometric Representation theory, University of Greenwich, UK, August 12-16, 2019.
- Workshop on Shimura varieties, representation theory and related topics, Hokkaido University, Japan, July 15-19, 2019.
- Hessenberg Varieties in Combinatorics, Geometry and Representation Theory, Banff International Research Station for Mathematical Innovation and Discovery, Canada, October 21-26, 2018.
- Future Directions in Representation Theory, University of Sydney, December 4-8, 2017.
- Nilpotent Orbits and Representation Theory, Scuola Normale Superiore, Pisa, Italy, June 13-16, 2016.
- Workshop on Springer Theory and Related Topics, University of Massachusetts, Amherst, USA, October 9-11, 2015.
- Conference on Geometry and Lie Theory, University of Hong Kong, March 5-8, 2014