Combinatorics of the Fock Space Representation of Uq (slp)
and its Crystal Basis
Chun-Ju Lai
April 19, 2012
Abstract
In 1989, Kashiwara introduced the notion of crystal basis for integrable Uq (g)-module, on
which the Kashiwara operators act nicely. He hi
Notes on Lusztigs Conjecture
Chun-Ju Lai
July 13, 2012
Abstract
This note serves as a summary to Fiebigs 8-hour lecture in the 2012 ECNU summer school
in Shanghai, China. He gave a partial proof of Lusztigs conjecture by proving an equivalent
conjecture b
Notes on Representations of
Semisimple Lie Algebras in the
BGG Category O
Author:
(Chun-Ju Lai)
[email protected]
NCTS/TPE
Supervisor:
(Shun-Jen Cheng)
Academia Sinica
Preface
I would give my gratitude to Professor Shun-Jen Cheng(). Under his supervi
Math 8852. Unitary representations and property (T ).
Problem Set 1.
Below [BHV] refers to the book Kazhdans property (T ) by Bekka, de la
Harpe and Valette.
1. Let G be a locally compact group.
(a) Prove that the left-regular representation (, L2 (G) is
Math 8852. Unitary representations and property (T ).
Problem Set 2.
Below [BHV] refers to the book Kazhdans property (T ) by Bekka, de la
Harpe and Valette.
1. Let H (Z) = x, y, z | z = [x, y ], [x, z ] = [y, z ] = 1 be the Heisenberg group
over Z. Let (
Math 8852. Unitary representations and property (T ).
Problem Set 4.
1. backlog: Problem 2 from Homework #3 (realizing irreducible representations of the Heisenberg group as induced representations)
2. backlog: Problem 3 from Homework #2 (outline of the p
Math 8852. Unitary representations and property (T ).
Problem Set 5.
1 (backlog from Homework #4). Let G be a locally compact abelian group,
and x G = Hom(G, S 1 ). Given > 0 and a compact subset K of G, let
U,K, be the set of G such that |(k ) (k )| < fo
Math 8852. Unitary representations and property (T ).
Problem Set 3.
Below [BHV] refers to the book Kazhdans property (T ) by Bekka, de la
Harpe and Valette.
1. (leftover from Problem Set 2). Let G = H (Z) = x, y, z | z = [x, y ], [x, z ] =
[y, z ] = 1 be