R440, Astronomy-Mathematics Building, NTU
Speaker(s):
Chih-Whi Chen (NCTS)
Organizer(s):
Chih-Whi Chen (NCTS)
Yung-Ning Peng (National Central University)
Chun-Ju Lai (Institute of Mathematics, Academia Sinica)
1. Course Background & Purposes
Kostant [1] initiated the study of Whittaker modules over finite-dimensional complex semisimple Lie algebras relative to non-singular characters ζ of the nil-radicals from triangular decompositions. A classification of simple Whittaker modules was obtained by Miliˇci´c and Soergel [3] based on the earlier work of McDowell [2]. This construction was obtained by means of certain standard Whittaker modules. Furthermore, Miliˇci´c and Soergel developed a category
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that contains these structural modules and specializes to a thick version of the BGG category
in the case when
=0.
There is an equivalence between
and certain singular category of Harish-Chandra bimodules established by Miliˇci´c and Soergel in [3]. Using this connection they proved that the composition series of the standard Whittaker modules associated to integral weights can be computed by means of Kazhdan-Lusztig polynomials. Around the same time, Backelin [4] completely solved this multiplicity problem using his exact functor Γ_ζ which sends modules in
to Whittaker modules in
.
Our goal is to introduce these classical results and to report on some recent progress of
Whittaker modules and categories for quasireductive Lie superalgebras.
2. Course Outline & Descriptions
Week 1: BGG category
We provide some background materials on the BGG category
for semisimple Lie algebras. In
particular, we will introduce the Kazhdan-Lusztig conjectures.
Week 2: Whittaker modules for Lie algebras.
We introduce some results of Kostant [1], McDowell [2], Miliciˇc -Soergel [3] and Backelin [4]. We will review the classification and construction of simple and standard Whittaker modules. We will review the multiplicity problem of standard Whittaker modules. In particular, we will introduce Backelin’s approach to this problem.
Week 3: Whittaker categories for Lie algebras.
For Week 3, we introduce Miliciˇc-Soergel’s approach [3] to the study of Whittaker category
and the multiplicity problem of standard Whittaker modules. We will explain the notion of cokernel categories in the sense of Bernstein-Gelfand and consider Harish-Chandra bimodules. We will investigate the structure of certain Whittaker categories.
Week 4: Whittaker modules and categories for Lie superalgebras, I.
We will introduce Lie superalgebras from Kac’s list and their categories
. We will mainly focus on Lie superalgebras of type I. We introduce the classification of simple and standard Whittaker modules [5]. We develop a super analogue of Backelin’s functor Γ_
and solve the multiplicity problem of standard Whittaker modules for Lie superalgebras of type I.
Week 5: Whittaker modules and categories for Lie superalgebras, II.
We introduce two theories of Harish-Chandra bimodules for Lie superalgebras. As an application, we develop a Miliciˇc-Soergel type equivalence [5] between certain Whittaker categories and Harish-Chandra bimodules. Use this equivalence, we obtain an alternative approach to the multiplicity problem and give a description of annihilator ideals of simple Whittaker modules and provide an equivalence between a certain Whittaker category and a certain category of modules over finite W-algebra for the general linear Lie superalgebra.
Contact:
murphyyu@ncts.ntu.edu.tw
