This thesis deals with a conjecture by Gordon James from 1990 and a
variant by Meinolf Geck about decomposition maps of the generic
Iwahori-Hecke algebra H of the symmetric group on n points arising
from specialization to characteristic l>0. If the parameter of the
algebra is specialized to an element q of GF(l) with multiplicative
order e, the conjecture states, that, for el>n, the decomposition map
does not depend on l and q, but only on e.

The main result of the present work is the following reformulation of
the James-Geck conjecture: For el>n every primitive idempotent in the
extension of scalars AH is primitive as idempotent in the algebra A'H,
where A and A' are rings of characteristic 0 (which are explicitly
constructed for l and q), such that A is contained in A' and therefore
AH in A'H. Thus an equivalent statement is given, which contains
only rings and algebras of characteristic 0. To prove equivalence,
generalizations of the well-known methods of lifting of idempotents and
of Brauer reciprocity are developed.

In addition results and observations are presented, that might lead to
an approach to a proof of the above mentioned reformulation. A method
is presented, how one can derive explicit formulae for primitive
idempotents in non-semi-simple symmetric algebras using matrix
representations on projective indecomposable modules. Such matrix
representations seem to arise naturally from the Kazhdan-Lusztig cell
modules.

Another possible attack for a proof stems from another result of the
present thesis, an explicit construction of a Wedderburn decomposition
for H using the Kazhdan-Lusztig basis.