中国科学院数学与系统科学研究院

数学研究所

学术报告

表示论研讨班

Speaker: Prof. Gurbir Dhillon（UCLA）

Title: Kostant slice and the principal Whittaker model

Time&Venue: 2024年12月11日（星期三） 14:30-15:30 & 南楼N820

Abstract: In the first week of this series, we will discuss the representation theory of finite W-algebras. The two most basic finite W-algebras, and not coincidentally the first ones studied historically, are those associated to the two extremal nilpotent orbits, namely the zero orbit and the principal orbit. For the zero orbit, one simply recovers the usual enveloping algebra of one’s Lie algebra. For the other extreme, namely the principal orbit, it is a beautiful theorem of Kostant that one obtains the center of the enveloping algebra. In this talk, we will review this story, its counterpart for finite groups of Lie type, i.e., Whittaker models, its semiclassical counterpart, i.e., the geometry of the locus of regular elements, and the relation to Soergel theory.

Title: Slodowy slice and the finite W-algebra

Time&Venue:  2024年12月12日（星期四） 14:30-15:30 & 南楼N818

Abstract: When one passes to all the orbits intermediate between the zero and principal orbits, one has analogues of the previous story. To explain this, we will first review some salient facts about the geometry of nilpotent orbits, particularly their classification and the crucial notion of a distinguished orbit. We will then describe the associated operation of taking a generalized Whittaker coefficient, as considered by Gelfand--Graev, Moeglin--Waldspurger, and many others. By considering the microlocal geometry of this construction, we will encounter the Slodowy slice, an affine space transverse to a given nilpotent orbit e, as well as its quantization, the finite W-algebra, following Harish-Chandra, Grothendieck, Slodowy, Premet, de Boer--Tjin, Gan--Ginzburg, Gaitsgory--Raskin, and many others.

Title: Category O for finite W-algebras and Springer theory

Time&Venue: 2024年12月13日（星期五） 10:30-11:30 & 南楼N818

Abstract: To each finite W-algebra, one has an associated Category O of highest weight representations introduced by Brundan--Goodwin--Kleschev, generalizing the work of Bernstein--Gelfand--Gelfand and Kostant in the case of the extremal nilpotent orbits. When e is principal in a Levi, the character formulas are well known, and go back to Kazhdan--Lusztig, Beilinson--Bernstein, Brylinski--Kashiwara, and Milicic--Soergel. For general nilpotents, the story contains several beautiful new complications. Some wonderful work of Losev gives the character formulas for certain central characters; we will explain this and some of its applications in Lie theory. We will then explain what the character formulas look like in general, based on work of Lusztig, Bezrukavnikov, Mirkovic, Losev, and ongoing work Arakawa--D.--Faergeman, and highlight in particular the relation to Springer fibers.

Title: Representation theory of the Virasoro Lie algebra

Time&Venue: 2024年12月16日（星期一） 14:30-15:30 & 南楼N818

Abstract: In the second week of the series, we will pass from studying finite W-algebras to their loop space analogues, i.e., the affine W-algebras. A completely fundamental example, for both intuition and applications, is the Virasoro algebra, which one can obtain from the principal orbit in sl_2. However, before developing this perspective, we will first describe the Virasoro algebra and its representation theory from `first principles’, following the fundamental works of Kac, Feigin, Fuchs, Frenkel, Wakimoto, and many others.

Title: Affine Lie algebras and affine W-algebras

Time&Venue: 2024年12月17日（星期二） 10:30-11:30 & 南楼N820

Abstract: Many features of the representation theory of the Virasoro algebra are most transparently understood via its presentation as the principal Drinfeld-Sokolov reduction of affine sl_2. In this talk, we will explain the basics of affine Lie algebras and the associated affine W-algebras, following the work of Kac, Feigin, Frenkel, Roan, Wakimoto, and many others. As a first application, we will revisit some of the phenomena witnessed for Virasoro from this perspective.

Title: Representation theory of affine W-algebras I

Time&Venue: 2024年12月18日（星期三） 10:30-11:30 & 南楼N818

Abstract: Just as we saw for finite W-algebras, the Category O for an affine W-algebra is most easily understood in the case of e a principal nilpotent in a Levi. We will first describe what the representation theory looks like in this case, following work of Kac, Lusztig, Deodhar, Gabber, Feigin, Frenkel, Wakimoto, Kashiwara, Tanisaki, Gaitsgory, Roan, Fiebig, Arakawa, Adamovic, Raskin, and many others, including some ongoing work of Arakawa--D.--Faergeman--Yang.

Title: Representation theory of affine W-algebras II

Time&Venue: 2024年12月19日（星期四） 14:30-15:30 & 南楼N818

Abstract: Having understood in the previous lecture what the Category O of an affine W-algebra looks like in the case of a principal nilpotent in a Levi, we will discuss in this lecture the story for a general nilpotent e, including the simple characters. This is based on ongoing work of Arakawa--D.--Faergeman. Time permitting, we will also sketch and speculate on some aspects of the wildly ramified representation theory, based on ongoing work of D.--Yang.

Title: Representation theory of affine W-algebras III

Time&Venue: 2024年12月20日（星期五） 10:30-11:30 & 南楼N818

Abstract: In the final talk of the series, we will discuss some more advanced topics, centered around on the ongoing interactions between affine W-algebras and the geometric Langlands program, including relative, quantum, and modular aspects. This is based on works and conjectures of Feigin, Frenkel, Beilinson, Drinfeld, Stoyanovsky, Gaitsgory, Arinkin, Fedorov, Bezrukavnikov, Braverman, Finkelberg, Mirkovic, Gaiotto, Rapcak, Ginzburg, Travkin, Yang, Creutzig, Linshaw, Nakatsuka, Raskin, D., Losev, and many others.