  This paper consists of three interconnected parts. Parts I,III study the
relationship between the cohomology of a reductive group and that of a Levi
subgroup. For example, we provide a necessary condition, arising from
Kazhdan-Lusztig theory, for the natural map on Ext-groups of irreducible
modules to be surjective. In cohomological degree 1, the map is always an
isomorphism, under our hypothesis. These results were inspired by recent work
of Hemmer obtained for general linear groups, and they both extend and improve
upon his work when our condition is met. Part II obtains results on Lusztig
character formulas for reductive groups, obtaining new necessary and sufficient
conditions for such formulas to hold. In the special case of general linear
groups, these conditions can be recast in a striking way completely in terms of
explicit representation theoretic properties of the symmetric group (and the
results improve upon the sufficient cohomological conditions established
recently in the authors).

Group Theory