代数群表示论第2版影印版_Jens Carsten Jantzen著.pdf

IntroductionI This book is meant to give its reader an introduction to the representationtheory of such groups as the general linear groups Gn(k),the special linear groupsSLn(k),the special orthogonal groups SOn(k),and the symplectic groups Sp2n(k)over an algebraically closed field k.These groups are algebraic groups,and we shalllook only at representations G-GL(V)that are homomorphisms of algebraicgroups.So any G-module(vector space with a representation of G)will be a spaceover the same ground field k.Many different techniques have been introduced into the theory,especiallyduring the last thirty years.Therefore,it is necessary(in my opinion)to start witha general introduction to the representation theory of algebraic group schemes.Thisis the aim of Part I of this book,whereas Part II then deals with the representationsof reductive groups.II The book begins with an introduction to schemes(Chapter I.1)and to(affine)group schemes and their representations(Chapter I.2).We adopt the functorialpoint of view for schemes.For example,the group scheme SLn over Z is thefunctor mapping each commutative ring A to the group SLn(A).Almost everythingabout these matters can also be found in the first two chapters of DG.I havetried to enable the reader to understand the basic definitions and constructionsindependently of DG.However,I refer to DG for some results that I feel thereader might be inclined to accept without going through the proof.Let me addthat the reader(of Part I)is supposed to have a reasonably good knowledge ofvarieties and algebraic groups.For example,he or she should know Bo up toChapter III,or the first seventeen chapters of Hu2,or the first six ones of Sp2.(There are additional prerequisites for Part II mentioned below.)In Chapter I.3,induction functors are defined in the context of group schemes,their elementary properties are proved,and they are used to construct injectivemodules and injective resolutions.These in turn are applied in Chapter I.4 to theconstruction of derived functors,especially to that of the Hochschild cohomologygroups and of the derived functors of induction.In contrast to the situation for finitegroups,the induction from a subgroup scheme H to the whole group scheme G is(usually)not exact,only left exact.The values of the derived functors of inductioncan also be interpreted(and are so in Chapter I.5)as cohomology groups of certainassociated bundles on the quotient G/H(at least for algebraic schemes over a field).Before doing that,we have to understand the construction of the quotient G/H.The situation gets simpler and has some additional features if H is normal in G.This is discussed in Chapter I.6.One can associate to any group scheme G an(associative)algebra Dist(G)ofdistributions on G(called the hyperalgebra of G by some authors).When workingover a field of characteristic 0,it is just the universal enveloping algebra of the Lievii