  We construct Nakajima's quiver varieties of type A in terms of conjugacy
classes of matrices and (non-Slodowy's) transverse slices naturally arising
from affine Grassmannians. In full generality quiver varieties are embedded
into Beilinson-Drinfeld Grassmannians of type A. Our construction provides a
compactification of Nakajima's quiver varieties and a decomposition of an
affine Grassmannian into a disjoint union of quiver varieties. As an
application we provide a geometric version of skew and symmetric $(GL(m),
GL(n))$ duality.
