Double centralizer properties play a central role in many parts of algebrai
c Lie theory. Soergel's double centralizer theorem relates the principal bl
ock of the Bernstein-Gelfand-Gelfand category O of a semisimple complex Lie
algebra with the coinvariant algebra (i.e., the cohomology algebra of the
corresponding flag manifold). Schur-Weyl duality relates the representation
theories of general linear and symmetric groups in defining characteristic
, or (via the quantized version) in nondefining characteristic. In this pap
er we exhibit algebraic structures behind these double centralizer properti
es. We show that the finite dimensional algebras relevant in this context h
ave dominant dimension at least two with respect to some projective-injecti
ve or tilting modules. General arguments which combine methods from ring th
eory (QF-3 rings and dominant dimension) with tools from representation the
ory (approximations, tilting modules) then imply the validity of these doub
le centralizer properties as well as new ones. In contrast to the tradition
al proofs (e.g., by the fundamental theorems of invariant theory) no comput
ations are necessary. (C) 2001 Academic Press.