Courses

  （1）Real Reductive Lie Groups and Discrete Series Representations by Professor Jing-Song Huang

  Abstract: The classification of discrete series representations by Harish-Chandra in 1960's was one of the greatest achievements of mathematics in 20th century. It laid down the foundation for the Langlands classification of admissible representations and the Langlands program. This mini-course is an introduction to Harish-Chandra theory of discrete series by using more recently developed techniques. We will begin with Schur's orthogonality relations for irreducible representations of compact Lie groups. The natural generalization is Harish-Chandra's orthogonality relations for discrete series of noncompact reductive Lie groups. We will also discuss the further generalization of Harish-Chandra's orthogonality relations to admissible representations.

  （2）Yangians of classical Lie algebras by Professor Naihuan Jing

  Abstract: The Yangian algebra was introduced by Drinfeld as a nontrivial example of noncommutative and noncocomutative Hopf algebra generalization of the enveloping algebra of a simple Lie algebra. First of part of the lectures will be devoted the Yangian of the general linear Lie algebra (type A). We will start with the first presentation using the R-matrix and go over some of its fundamental properties, then we will discuss the other two presentations and their identification. In the second part we will introduce the twisted Yangians as special subalgebras of the Yangian in type A to realize other types (BCD types) of Yangians and their presentations and possibly some basics of their representations.

  （3）Nilpotent orbits, primitive ideals and finite W-algebras by Professor Alexander Premet.

  Abstract: I will discuss several topics of modular representation theory of restricted Lie algebras.
  This will include the following topics:
    1. reduced enveloping algebras and the current state of the first Kac-Weisfeiler conjecture;
    2. nilpotent orbits and sheets in reductive Lie algebras;
    3. support varieties for non-restricted modules and the second Kac-Weisfeiler conjecture;
    4. finite W-algebras and their modular analogues;
    5. small modular representations and multiplicity-free primitive ideals;
    6. some open problems.

  （4）Quantum groups and quiver representations by Professor Weiqiang Wang

  Abstract: The goal of this mini-course is to provide an introduction to some basic structures of quantum groups from the viewpoint of quiver representations. Quantum groups are introduced by Drinfeld and Jimbo in 1980's as a deformation of the universal enveloping algebras of simple Lie algebras. A quantum group admits automorphisms (due to Lusztig) which satisfy the braid group relations for the associated Weyl group.
  Dynkin quivers are ADE Dynkin diagrams equipped with orientations. In this mini-course we study representations of a Dynkin quiver (equivalently, representations of a so-called path algebra), and classify its indecomposables via positive roots of the corresponding simple Lie algebra. By studying the quiver representations over finite fields, we formulate the notion of Ringel-Hall algebra and show that it provides a realization of a positive half of a quantum group, U^+. The reflection functors are formulated, which provide a construction of the braid group action on the quantum group. We shall describe PBW bases for U^+ (as well as monomial bases, and canonical basis, if time permitting).