 
Summary: THE STRONG LINKAGE PRINCIPLE FOR QUANTUM
GROUPS AT ROOTS OF 1
HENNING HAAHR ANDERSEN
Dedicated to R. Steinberg on the occasion of his 80'th birthday
Let G be a semisimple algebraic group over a eld of characteristic
p > 0 and let B 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 Gmodules breaks up into blocks corresponding
to the weight orbits of the aĆne 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 eld k. The parameter q is a nonzero 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 below), and we deduce that the category of
integrable U q modules splits up into blocks for an aĆne Weyl group
associated to U q (when q is not a root of unity it is well known and
