IntroductionLet ~r denote the local ring Z[v]~, where v is an indeterminate and ~ is the maximal ideal in Z[v] generated by v -1 and a fixed odd prime p. The residue field ~r = Fv is denoted by k. To each Cartan matrix (a~j)7,i= 1 Drinfeld [Dr] and Jimbo [Ji] have associated a so-called quantum group U', which is a Hopf algebra over Q(v) defined by certain generators and relations. Following Lusztig [L 5, L 6] we consider an ~r U of U' which is a Hopf algebra over d, and also the "specializations" Ur = U | F for various d-algebras F.Firstly we introduce the coordinate algebra ~r [ U] as a suitable dual of U. Our first main result says that d [ U] is a free ~r (Theorem 1.33). This relies on the connection, established in [loc. cit.], betweeen Uk and the hyperalgebra of the semi-simple algebraic group G k corresponding to (aij). Here k is made into an ~r by sending v to 1. The point is--and this will be used repeatedly throughout the paper--that this connection allows us to carry over information from the representation theory of Gk to that of Uk.Next we use the coordinate algebra to set up a general theory of induction. A crucial result here is that induction from the trivial subalgebra as well as from U 0 (see Section 0 for notations) is exact, see Theorem 1.31 and Proposition 2.11. Also, we emphasize the study of induction from "generalized parabolic subalgebras". We check that our induction functors have the standard properties, e.g. Frobenius reciprocity, transitivity and the tensor identity (Section 2). Moreover, we study their behaviour under base change, thereby getting explicit connections to the analogous functors in the representation theory of Gk and GQ, see Section 3.The above results together with a detailed examination of the rank 1 ease (Section 4) then enable us to obtain some deeper results about induction from a "Borel subalgebra". These include analogues of Serre's theorem, Grothendieek's theorem, Kempf's vanishing theorem for dominant characters and Demazure's character formula. Moreover, we show that the concepts and results about good, respectively excellent filtrations carry over to the quantum ease, see Section 5. 2 H.H. Andersen et al.Consider now a specialization of d into a field F. We develop a Borel-Weil-Bott theory for Ur, see Section 6. If the image ( ofo is not a root of 1 then the theory is completely analogous to the classical theory for G~ (regardless of the characteristic of F) whereas if char(F) = 0 and ( is a root of 1 then we have a situation resembling the modular representation theory for Gk. This latter situation is explored further in Section 8 where we prove a linkage principle and a translation principle for Ur. An important ingredient in the arguments there is Serre duality (Theorem 7.3) which in turn requires a special case of Bott's theorem.Everything has now been set up in a way which invites us to define a "Jantzen type" titration and prove a sum formula. In fact, we obtain several such filtrations and corresponding sum formulas (see Section 10). Working over k t...

No abstract

No abstract