TY - BOOK ID - 915403 TI - The discrete series of GLn over a finite field PY - 1974 VL - 81 SN - 0691081549 1400881765 9780691081540 PB - Princeton : Princeton University Press, DB - UniCat KW - Group theory KW - Algebraic fields KW - Linear algebraic groups KW - Representations of groups KW - Series KW - 511.33 KW - Algebra KW - Mathematics KW - Processes, Infinite KW - Sequences (Mathematics) KW - Group representation (Mathematics) KW - Groups, Representation theory of KW - Algebraic groups, Linear KW - Geometry, Algebraic KW - Algebraic varieties KW - Algebraic number fields KW - Algebraic numbers KW - Fields, Algebraic KW - Algebra, Abstract KW - Algebraic number theory KW - Rings (Algebra) KW - Analytical and multiplicative number theory. Asymptotics. Sieves etc. KW - Algebraic fields. KW - Linear algebraic groups. KW - Representations of groups. KW - Series. KW - 511.33 Analytical and multiplicative number theory. Asymptotics. Sieves etc. KW - Analytical and multiplicative number theory. Asymptotics. Sieves etc KW - Addition. KW - Affine group. KW - Automorphism. KW - Dimension. KW - Eigenvalues and eigenvectors. KW - Endomorphism. KW - Field of fractions. KW - Finite field. KW - Free module. KW - Grothendieck group. KW - Homomorphism. KW - Linear subspace. KW - Morphism. KW - P-adic number. KW - Partially ordered set. KW - Simplicial complex. KW - Tensor product. KW - Theorem. KW - Witt vector. KW - Groupes algébriques linéaires KW - Groupes algébriques linéaires KW - Représentations de groupes UR - https://www.unicat.be/uniCat?func=search&query=sysid:915403 AB - 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). ER -