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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.

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 GL_n (F_q), where V is the n-dimensional F-vector space on which GL_n(F_q) acts. This is a p-adic representation; more precisely D(V) is a free module of rank (q--1) (q²--1)...(q^n-1--1) over the ring of Witt vectors W_F of F.

In Chapter 1 the author...

According to Drinfeld, a quantum group is the same as a Hopf algebra. This includes as special cases, the algebra of regular functions on an algebraic group and the enveloping algebra of a semisimple Lie algebra. The qu- tum groups discussed in this book are the quantized enveloping algebras introduced by Drinfeld and Jimbo in 1985, or variations thereof. Although such quantum groups appeared in connection with problems in statistical mechanics and are closely related to conformal field theory and knot theory, we will regard them purely as a new development in Lie theory. Their place in Lie...