I This book is meant to give its reader an introduction to the representation
theory of such groups as the general linear groups GLn(k), the special linear groups
SLn(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 shall
look only at representations G GL(V) that are homomorphisms of algebraic
groups. So any G-module (vector space with a representation of G) will be a space
over the same ground field k.
Many different techniques have been introduced into the theory, especially
during the last thirty years. Therefore, it is necessary (in my opinion) to start with
a general introduction to the representation theory of algebraic group schemes. This
is the aim of Part I of this book, whereas Part II then deals with the representations
of reductive groups.
II The book begins with an introduction to schemes (Chapter 1.1) and to (affine)
group schemes and their representations (Chapter 1.2). We adopt the "functorial"
point of view for schemes. For example, the group scheme SLn over Z is the
functor mapping each commutative ring A to the group SLn(A). Almost everything
about these matters can also be found in the first two chapters of [DG]. I have
tried to enable the reader to understand the basic definitions and constructions
independently of [DG]. However, I refer to [DG] for some results that I feel the
reader might be inclined to accept without going through the proof. Let me add
that the reader (of Part I) is supposed to have a reasonably good knowledge of
varieties and algebraic groups. For example, he or she should know [Bo] up to
Chapter 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 1.3, induction functors are defined in the context of group schemes,
their elementary properties are proved, and they are used to construct injective
modules and injective resolutions. These in turn are applied in Chapter 1.4 to the
construction of derived functors, especially to that of the Hochschild cohomology
groups and of the derived functors of induction. In contrast to the situation for finite
groups, 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 induction
can also be interpreted (and are so in Chapter 1.5) as cohomology groups of certain
associated 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 1.6.
One can associate to any group scheme G an (associative) algebra Dist(G?) of
distributions on G (called the hyperalgebra of G by some authors). When working
over a field of characteristic 0, it is just the universal enveloping algebra of the Lie
Previous Page Next Page