## 2022 天元表示论讨论班

我们计划在本学期（5月-7月）的每周五下午三点邀请专家同行就约化群表示论中的一些前沿的技术做系列专题报告。

我们计划请相关领域的专家就特定专题，分2到3次，详细阐述其中的关键概念和技术，以促进同行们互相交流合作。每次报告时长约为1-1.5小时，欢迎大家参加，并在报告过程中参与讨论和提问。

报告形式：线上报告

腾讯会议ID：436 5742 3564（无密码）

Date | Time | Speaker | Title | Abstract | Host |
---|---|---|---|---|---|

2022/05/06 | 15:00-16:30 | 陈阳洋 | Schwartz homologies of Nash groups, part I | In this talk, we will give a brief introduction to the Schwartz induction theory of almost linear Nash groups. | Poster@XMU |

2022/05/13 | 15:00-16:30 | 陈阳洋 | Schwartz homologies of Nash groups, part II | Same as the above. | Poster@XMU |

2022/05/20 | 15:00-16:30 | 陈阳洋 | Schwartz homologies of Nash groups, part III | Same as the above. | XMU |

2022/05/27 | 15:00-16:30 | 陈哲 | Higher Deligne-Lusztig theory I | (I) In the first talk I would like to give an introduction to Deligne–Lusztig theory of reductive groups over finite fields, with a focus on the case of SL_2(F_q). Time permitting, I will also discuss some basics of the generalisation for reductive groups over discrete valuation rings (called higher Deligne–Lusztig theory). | XMU |

2022/06/03 | 15:00-16:30 | 陈哲 | Higher Deligne-Lusztig theory II | (II) In the second talk, I would like to discuss the algebraisation problem of higher Deligne–Lusztig representations raised by Lusztig, which seeks algebraic realisations (via, say, Clifford theory) of these geometrically constructed representations. I will discuss our resolution of this problem at even levels in a joint work with Stasinski in 2017, as well as our recent progress towards the odd level case. | XMU |

2022/06/10 | 15:00-16:30 | 陈哲 | Higher Deligne-Lusztig theory III | (III) In the third talk, I plan to discuss a curious restriction-to-torus formula of Deligne–Lusztig characters, which is motivated by a phenomenon appeared in the algebraisation problem and by a work of Reeder. | XMU |

2022/06/17 | 15:00-16:30 | 聂思安 | Lusztig’s map from conjugacy classes of the Weyl group to the unipotent classes | For a connected reductive group over an algebraically closed field, Lusztig constructed a miraculous map from the conjugacy classes of the Weyl group to the unipotent conjugacy classes. We will discuss various interesting properties and applications of the map. Part of the talk is based on joint work with Jeffrey Adams and Xuhua He. | XMU |

2022/06/24 | 15:00-16:30 | 聂思安 | Counting points on Newton strata and a multiplicity-one phenomenon | Affine Deligne-Lusztig varieties play an important role in the theory of Shimura varieties. Their geometries are controlled by the Newton strata in Iwahori double cosets. In this talk, I will report a multiplicity-one phenomenon on the Newton strata of a large class of Iwahori double cosets, namely, these strata form a stratification and each of them is nonempty and irreducible. We prove it by counting rational points on the Newton strata. As a consequence, we show the corresponding affine Deligne-Lusztig varieties are explicit unions of classical Deligne-Lusztig varieties of Coxeter type. This is based on joint work in progress with X. He and Q. Yu. | XMU |

## 2020 Fall Seminar on real reductive groups

This semester, 2020 Fall, we will focus on the representation theory of real reductive Lie groups.

Throughout these seminar talks, we hope to clarify the current status of the subject and stimulate collaborations. We expect the speakers to explain the technical details. The audiences are encouraged to interrupt the speaker and raise questions/make comments. Most lectures will be given in Chinese.

Click the title for the videos of the talks.

Time | Conference Id | Speaker | Title | Abstract | Host |
---|---|---|---|---|---|

2020/09/02 | NA | 董超平 | Towards the classification of Dirac series for real classical groups | Prof. Dong will collect the math ingredients pertaining to the classification of Dirac series. In particular, he will mention its relation with unipotent representations. | SJTU |

2020/09/25 2pm-4pm | NA | 白占强 | Gelfand-Kirillov dimensions and associated varieties of highest weight modules | In this talk, I will give some introduction to Gelfand-Kirillov dimensions and associated varieties of highest weight modules of Lie algebras (groups). Then I will talk about our work on GKdim and Associated Varieties of highest weight (Harish-Chandra) modules. | SJTU |

2020/10/09 2pm-3pm | 白占强 | Gelfand-Kirillov dimensions and associated varieties of highest weight modules II | See above | SJTU | |

2020/10/09 3:30pm-4:30pm | 余世霖 | Geometric quantization I | The speaker will introduce the notion of Poisson algebra and deformation quantization | SJTU | |

2020/10/16 2:00pm-3:00pm | 腾讯会议 ID：864 4848 4933 会议密码：1122 | 余世霖 | Geometric quantization II | The speaker will continue his talk last week on the notion of Poisson algebra and deformation quantization. Then he will discuss some results on the quantization of a nilpotent coadjoint orbit of a real reductive Lie group. | SJTU |

2020/10/16 3:30pm-4:30pm | see above | 马家骏 | Associated character formula in Howe correspondence I | I will discuss the joint works with Loke and Barbasch-Sun-Zhu on the formula of the associated character of a local theta lifting (Howe correspondence). Furthermore, I will explain its applications in the construction of unipotent representations and geometric quantization. | XMU |

2020/10/23 2:00pm-3:00pm | see above | 马家骏 | Associated character formula in Howe correspondence II | see above | see above |

2020/10/23 3:30pm-4:30pm | see above | Daniel Wong | On NON-unitary (g,K)-modules for complex simple Lie groups I | We discuss several techniques detecting non-unitarity of representations of complex Lie groups G, such as bottom layer K-types, Dirac inequality and deformation of parameters. These techniques are essential in determining the unitary spectrum of G. We will also briefly mention where the difficulties arise as one applies the same techniques for general real reductive groups. | SJTU |

2020/10/30 2:00pm-3:00pm | Zoom: 61208845721 pass:130094 | Daniel Wong | On NON-unitary (g,K)-modules for complex simple Lie groups II | See above | SJTU |

2020/10/30 3:30pm-4:30pm | Zoom: 61208845721 pass:130094 | Li Ning | On certain invariants for constituents of degenerate principal series of Sp(2n,R) | In this talk, I will describe associated cycles and wave front cycles for irreducible constituents of degenerate principal series of Sp(2n,R), and their relationship with the space of generalized Whittaker models of these representations. This can be achieved by a careful study of Loke-Ma and Gomez-Zhu’s work. | SJTU |

## 2018-Spring Representations of Finite group and affine Hecke algebra

For 2018 Spring semester, from 28 Feb 2018 to the end of the semester.

**Time:** Wednesday afternoon **13:00-15:30**, Week 1-17.

**Venue:** School of Mathematics SJTU, Room-1106.

We plan to read the following two papers.

1. Pierre Deligne and George Lusztig, Representations of Reductive Groups Over Finite Fields

2. David Kazhdan, George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras

The tasks are divied as the following:

Speackers | Deligne-Lusztig’s paper |
---|---|

Ma Jiajun | Introduction, Section 1-2 |

Chen Miaofen | Section 3-4 |

We managed to cover Deligne-Lusztig’s paper in that semester.

## 2017-Fall Minimal length elements

For 2017 Fall semester, from 11-Sep 2017 to 21-Nov 2017, the time and venue are as the following:

**Time:** Wednesday afternoon **13:00-15:00**.

**Venue:** School of Mathematics SJTU, Room-1106.

Time/speaker | Venue | Title/Abstract |
---|---|---|

Sep 13, 2017, 13:00-15:00 | Room-1106 | Introduction on conjugacy classes of finite coxeter groups. |

Ma Jia-Jun | We will discuss the geometric interpretation of the length function of finite Coxeter groups. | |

Sep 20, 2017, 13:00-15:00/Ma Jia-Jun | Room-1106 | Geometric interpretation of the length function. (cont.) |

Sep 27, 2017, 14:00-15:00/Ma Jia-Jun | Room-1106 | Proof of the main theorem on minimal length elements (for finite Coxeter group) |

Oct 4, 2017 | Cancelled due to the national day holiday. | |

Oct 11, 2017 | Cancelled due to an event of the department. | |

Oct 18, 13:00-15:00/Zhang Guanglian | Room-1106 | Elliptic elements in Weyl groups |

Oct 25, 13:00-15:00/Zhang Guanglian | Room-1106 | Good elements in Weyl groups |

Nov 1, 13:00-15:00/Qin Fan | Room-1106 | Introduction to affine Weyl groups. |

Nov 8, 13:00-15:00/Qin Fan | Room-1106 | minimal length element in affine groups. |

Nov 15, 13:00-15:00/Qin Fan | Room-1106 | minimal length element in affine groups (cont.) |

Nov 22 | Cancelled | |

Nov 29 13:00-15:00/Chen Miaofen | Room-1106 | Straight conjugacy class |

Dec 6 13:00-15:00/Chen Miaofen | Room-1106 | Straight conjugacy class and its relation to other topics. |

Dec 13: 13:00-15:00/ Ma Jia-Jun | Room-1106 | Classification of nice elements. |

Dec 20 10:00-12:00/ Ma Jia-Jun | Room-1106 | Cocenter of an affine Hecke algebra. |

## 2017-Spring Springer theory

For 2017 Spring semester, from 21-Mar 2017 to 27-June 2017, the time and venue are as the following:

**Time:** Tuesday noon **12:00-13:30**. (the time slot is updated)

**Venue:** School of Mathematics SJTU, Room-1106.

Time | Venue | Title/Abstract |
---|---|---|

Mar 21, 2017, 18:30-20:00 | Room-1106 | Introduction to Springer theory |

This is the first talk of my proposed seminar on Weyl group, Springer theory, Hecke algebra and related topics. As an orientation, I will first explain the content of Springer correspondence. Then I will briefly review the basic idea of the constructions of the Springer correspondence. In the end, I will discuss some relationships of the Springer theory with other fields, such as combinatorics and representation theory of reductive groups. If time permits, I will also discuss the generalized Springer correspondence. | ||

Mar 28, 2017 | Canceled | |

April 4, 2017 | Canceled due to Qingming Festival | |

April 11, 2017, 12:00-13:30 | Room-1106 | A proof of Springer correspondence by Chriss-Ginzburg, (I) |

I will present a proof of the Springer correspondence following Chriss-Ginzburg's book "Representation Theory and Complex Geometry" in the next two or three talks. In this talk, I plan to discuss the geometry of the flag variety and the Steinberg variety. | ||

April 18, 2017, 12:00-13:30 | Room-1106 | |

Continue the discussion on Ghriss-Ginzburg's book. | ||

April 25, 2017, 12:00-13:30 | Room-1106 | |

Continue the discussion on Ghriss-Ginzburg's book. | ||

May 2 , 2017, 12:00-13:30 | Canceled | |

May 9 , 2017, 12:00-13:30 | Room-1106 | Discussion on Ghriss-Ginzburg's book: Borel-Moore homology |

May 16 , 2017, 12:00-13:30 | Room-1106 | Discussion on Ghriss-Ginzburg's book: Borel-Moore homology |

May 23 , 2017, 12:00-13:30 | Room-1106 | Discussion on Ghriss-Ginzburg's book. |

May 30 , 2017, 12:00-13:30 | Canceled due to Duanwu Festival(Dragon Boat Festival) | |

Jun 6 , 2017, 12:00-13:30 | Discussion on Ghriss-Ginzburg's book. | |

Jun 13 , 2017, 12:00-13:30 | Discussion on Ghriss-Ginzburg's book. | |

Jun 20 , 2017, 12:00-13:30 | Discussion on Ghriss-Ginzburg's book. Finished Chapter 3 |