Geometric Representation Seminar
Student No.:40
Time:Fri 15:20-16:20, Oct.12-Jan.18
Instructor:Shan Peng  
Place:Conference Room 3, Jin Chun Yuan West Bldg.
Starting Date:2018-10-12
Ending Date:2019-1-18


Speaker: 何旭华 He Xuhua (University of Maryland)

Title: B(G) and beyond

Abstract: Let F be a nonarchimedean local field and F˘ be the completion of the maximal unramified extension of F. Let G be a connected reductive group over F. The set B(G) of the Frobenius-twisted conjugacy classes of G(F˘) is classified by Kottwitz. Motivated by Kottwitz’s work, I introduced a decomposition of G(F) into certain conjugate-invariant subsets, which are called the Newton strata. This decomposition has then found applications in the study of representations of p-adic groups, e.g. the Howe's conjecture and the trace Paley-Wiener theorem in the mod-l setting. In this talk, I will explain the relation between the Newton maps over F and over F˘, and the relation between B(G) and the Newton decomposition of G(F). This is based on a recent joint work with S. Nie.


Speaker: 余君 Yu Jun (BICMR)

Title: A geometric interpretation of Kirillov's conjecture

Abstract: Kirillov's conjecture says that the restriction to a microbolic subgroup of any irreducible unitary representation of GL(n,k) with k an archimedean local field or a p-adic local field is still irreducible. This is proved to be true by Bernstein for p-adic local fields and by Sahi, Sahi-Stein and Baruch for archimedean local fields. In this talk we present a geometric interpretation of Kirillov's conjecture from the point of view of orbit method. With our geometric study, we are able to show the Duflo's conjecture in this case. This is a joint work with Gang Liu.


Speaker: 覃帆 Qin Fan (上海交通大学)

Title: Generic bases of cluster algebras

Abstract: It is natural to expect that the cluster characters of generic objects in a cluster category often provide a basis for the corresponding (upper) cluster algebra, called the generic basis. Previously known cases are cluster algebras arising from coordinate rings of unipotent cells or triangulated surfaces.
In this talk, we first review the construction of generic bases. Then, we present a recent result that all injective-reachable upper cluster algebras possess generic bases. It covers all previously known cases and many more. It applies to many cluster algebras with a Lie theoretic background.


Speaker: 胡峻 Hu Jun (北京理工大学)

Title: Grassmannian, symmetric functions and cyclotomic nilHecke algebras

Abstract: The nilHecke algebras and their cyclotomic quotients are closely related to the categorification of quantum $sl_2$ and the geometry of Grassmannian. In this talk, I shall present a purely algebraic approach to the study of the cyclotomic nilHecke algebras. Each Schubert class basis element of the cohomology of the Grassmannian $G_{n,\ell-n}$ will be identified with certain purely algebraically defined integral basis element of the $Z$-graded basic algebra of a cyclotomic nilHecke algebra. The center of the latter will be shown to have a basis given by the evaluation of certain Schur (symmetric) polynomial at $y_1,\cdots,y_n$. We will also give an explicit description of Shan-Varagnolo-Vasserot's trace form on these algebras.


Speaker: 胡永泉 Yongquan Hu (Chinese Academy of Sciences)

Title: Dimensions of Bianchi modular forms and mod p representations of GL2(Qp)

Abstract: Given a level $N$ and a weight $k$, we know the dimension of the space of (classical) modular forms. This turns out to be unknown if we consider Bianchi modular forms, that is, modular forms over imaginary quadratic fields. In this talk, we consider the asymptotic behavior of the dimension when the level is fixed and the weight grows. I will first explain an upper bound obtained by Marshall using Emerton's completed cohomology and the theory of Iwasawa algebras. Then I will review the mod p representation theory of GL2(Qp) and explain how to use it to improve this bound.