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The quantum groups discussed in this book are the quantized
enveloping algebras introduced by Drinfeld and Jimbo in 1985, or
variations thereof. The theory of quantum groups has led to a new,
extremely rigid structure, in which the objects of the theory are
provided with canonical basis with rather remarkable properties.
This book will be of interest to mathematicians working in the
representation theory of Lie groups and Lie algebras, knot
theorists and to theoretical physicists and graduate students.
Since large parts of the book are independent of the theory of
perverse sheaves, the book could also be used as a text book.
Two basic problems of representation theory are to classify irreducible representations and decompose representations occuring naturally in some other context. Algebras of Iwahori-Hecke type are one of the tools and were, probably, first considered in the context of representation theory of finite groups of Lie type. This volume consists of notes of the courses on Iwahori-Hecke algebras and their representation theory, given during the CIME summer school which took place in 1999 in Martina Franca, Italy.
In this book Professor Lusztig solves an interesting problem by
entirely new methods: specifically, the use of cohomology of
buildings and related complexes. The book gives an explicit
construction of one distinguished member, D(V), of the discrete
series of GLn (Fq), where V is the n-dimensional F-vector space on
which GLn(Fq) acts. This is a p-adic representation; more precisely
D(V) is a free module of rank (q--1) (q2--1)...(qn-1--1) over the
ring of Witt vectors WF of F. In Chapter 1 the author studies the
homology of partially ordered sets, and proves some vanishing
theorems for the homology of some partially ordered sets associated
to geometric structures. Chapter 2 is a study of the representation
? of the affine group over a finite field. In Chapter 3 D(V) is
defined, and its restriction to parabolic subgroups is determined.
In Chapter 4 the author computes the character of D(V), and shows
how to obtain other members of the discrete series by applying
Galois automorphisms to D(V). Applications are in Chapter 5. As one
of the main applications of his study the author gives a precise
analysis of a Brauer lifting of the standard representation of
GLn(Fq).
This book presents a classification of all (complex) irreducible
representations of a reductive group with connected centre, over a
finite field. To achieve this, the author uses etale intersection
cohomology, and detailed information on representations of Weyl
groups.
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