  The degenerate affine and affine BMW algebras arise naturally in the context
of Schur-Weyl duality for orthogonal and symplectic Lie algebras and quantum
groups, respectively. Cyclotomic BMW algebras, affine Hecke algebras,
cyclotomic Hecke algebras, and their degenerate versions are quotients. In this
paper the theory is unified by treating the orthogonal and symplectic cases
simultaneously; we make an exact parallel between the degenerate affine and
affine cases via a new algebra which takes the role of the affine braid group
for the degenerate setting. A main result of this paper is an identification of
the centers of the affine and degenerate affine BMW algebras in terms of rings
of symmetric functions which satisfy a "cancellation property" or "wheel
condition" (in the degenerate case, a reformulation of a result of Nazarov).
Miraculously, these same rings also arise in Schubert calculus, as the
cohomology and K-theory of isotropic Grassmanians and symplectic loop
Grassmanians. We also establish new intertwiner-like identities which, when
projected to the center, produce the recursions for central elements given
previously by Nazarov for degenerate affine BMW algebras, and by
Beliakova-Blanchet for affine BMW algebras.
