Dedicated to R. Steinberg on the occasion of his 80th birthday
Let G be a semisimple algebraic group over a field of characteristic p > 0 and let B β G be a Borel subgroup. The strong linkage principle for G in [1] gives a condition on the highest weights of composition factors of the cohomology of line bundles on G/B. As a consequence the category of rational G-modules breaks up into blocks corresponding to the weight orbits of the affine Weyl group associated with G.
In this paper we shall discuss the same principle for quantum groups. More precisely, we let R denote the root system for G relative to the maximal torus T in B, and we denote by U q the corresponding quantum group over a field k. The parameter q is a non-zero element of k and U q is the specialization of Lusztig's Z[v, v -1 ]-form of the usual quantized enveloping algebra over Q(v), see [5]. When q is a root of unity we prove an analogous strong linkage principle for certain cohomology modules for U q (see Theorem 3.1), and we deduce that the category of integrable U q -modules splits up into blocks for an affine Weyl group associated to U q (when q is not a root of unity it is well known and easy to check that this category is semisimple).
The quantum version of the strong linkage principle presented in this paper is not new. In the case where the characteristic of k is 0 and the order of q is an odd prime power it appears in [5]. More general versions are found in [4,6,10]. Our treatment will be uniform covering all fields k and roots of unity of arbitrary order. Moreover, our proof is elementary and almost self-contained (taking only standard facts about representations of U q for granted and leaving some of the details of the sl 2 -case to the reader).
As a byproduct of our proof we get the finite dimensionality of the cohomology modules we are dealing with. In addition to the above mentioned application toward the splitting