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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).

Group theory --- Algebraic fields --- Linear algebraic groups --- Representations of groups --- Series --- 511.33 --- Algebra --- Mathematics --- Processes, Infinite --- Sequences (Mathematics) --- Group representation (Mathematics) --- Groups, Representation theory of --- Algebraic groups, Linear --- Geometry, Algebraic --- Algebraic varieties --- Algebraic number fields --- Algebraic numbers --- Fields, Algebraic --- Algebra, Abstract --- Algebraic number theory --- Rings (Algebra) --- Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- Algebraic fields. --- Linear algebraic groups. --- Representations of groups. --- Series. --- 511.33 Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- Analytical and multiplicative number theory. Asymptotics. Sieves etc --- Addition. --- Affine group. --- Automorphism. --- Dimension. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Field of fractions. --- Finite field. --- Free module. --- Grothendieck group. --- Homomorphism. --- Linear subspace. --- Morphism. --- P-adic number. --- Partially ordered set. --- Simplicial complex. --- Tensor product. --- Theorem. --- Witt vector. --- Groupes algébriques linéaires --- Groupes algébriques linéaires --- Représentations de groupes

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This book presents a coherent account of the current status of etale homotopy theory, a topological theory introduced into abstract algebraic geometry by M. Artin and B. Mazur. Eric M. Friedlander presents many of his own applications of this theory to algebraic topology, finite Chevalley groups, and algebraic geometry. Of particular interest are the discussions concerning the Adams Conjecture, K-theories of finite fields, and Poincare duality. Because these applications have required repeated modifications of the original formulation of etale homotopy theory, the author provides a new treatment of the foundations which is more general and more precise than previous versions.One purpose of this book is to offer the basic techniques and results of etale homotopy theory to topologists and algebraic geometers who may then apply the theory in their own work. With a view to such future applications, the author has introduced a number of new constructions (function complexes, relative homology and cohomology, generalized cohomology) which have immediately proved applicable to algebraic K-theory.

Algebraic topology --- 512.73 --- 515.14 --- Homology theory --- Homotopy theory --- Schemes (Algebraic geometry) --- Geometry, Algebraic --- Deformations, Continuous --- Topology --- Cohomology theory --- Contrahomology theory --- Cohomology theory of algebraic varieties and schemes --- 515.14 Algebraic topology --- 512.73 Cohomology theory of algebraic varieties and schemes --- Homotopy theory. --- Homology theory. --- Abelian group. --- Adams operation. --- Adjoint functors. --- Alexander Grothendieck. --- Algebraic K-theory. --- Algebraic closure. --- Algebraic geometry. --- Algebraic group. --- Algebraic number theory. --- Algebraic structure. --- Algebraic topology (object). --- Algebraic topology. --- Algebraic variety. --- Algebraically closed field. --- Automorphism. --- Base change. --- Cap product. --- Cartesian product. --- Closed immersion. --- Codimension. --- Coefficient. --- Cohomology. --- Comparison theorem. --- Complex number. --- Complex vector bundle. --- Connected component (graph theory). --- Connected space. --- Coprime integers. --- Corollary. --- Covering space. --- Derived functor. --- Dimension (vector space). --- Disjoint union. --- Embedding. --- Existence theorem. --- Ext functor. --- Exterior algebra. --- Fiber bundle. --- Fibration. --- Finite field. --- Finite group. --- Free group. --- Functor. --- Fundamental group. --- Galois cohomology. --- Galois extension. --- Geometry. --- Grothendieck topology. --- Homogeneous space. --- Homological algebra. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Integral domain. --- Intersection (set theory). --- Inverse limit. --- Inverse system. --- K-theory. --- Leray spectral sequence. --- Lie group. --- Local ring. --- Mapping cylinder. --- Natural number. --- Natural transformation. --- Neighbourhood (mathematics). --- Newton polynomial. --- Noetherian ring. --- Open set. --- Opposite category. --- Pointed set. --- Presheaf (category theory). --- Reductive group. --- Regular local ring. --- Relative homology. --- Residue field. --- Riemann surface. --- Root of unity. --- Serre spectral sequence. --- Shape theory (mathematics). --- Sheaf (mathematics). --- Sheaf cohomology. --- Sheaf of spectra. --- Simplex. --- Simplicial set. --- Special case. --- Spectral sequence. --- Surjective function. --- Theorem. --- Topological K-theory. --- Topological space. --- Topology. --- Tubular neighborhood. --- Vector bundle. --- Weak equivalence (homotopy theory). --- Weil conjectures. --- Weyl group. --- Witt vector. --- Zariski topology. --- Homologie --- Topologie algebrique --- Geometrie algebrique --- Homotopie