  We study a class of varieties which generalize the classical orbital
varieties of Joseph. We show that our generalized orbital varieties are the
irreducible components of a Mirkovic-Vybornov slice to a nilpotent orbit, and
can be labeled by semistandard Young tableaux. Furthermore, we prove that
Mirkovic-Vilonen cycles are obtained by applying the Mirkovic-Vybornov
isomorphism to generalized orbital varieties and taking a projective closure,
refining Mirkovic and Vybornov's result. As a consequence, we are able to use
the Lusztig datum of a Mirkovic--Vilonen cycle to determine the tableau
labeling the generalized orbital variety which maps to it, and, hence, the
ideal of the generalized orbital variety itself. By homogenizing we obtain
equations for the cycle we started with, which is useful for computing various
equivariant invariants such as equivariant multiplicity and multidegree. As an
application, we show that in type $A_5$ the Mirkovic-Vilonen basis and
Lusztig's dual semicanonical basis are not equal as perfect bases of the
coordinate ring of the unipotent subgroup. This is significant because it shows
that perfect bases are not unique. Our comparison relies heavily on the theory
of measures developed in Baumann, Kamnitzer, and Knutson's MV Basis and DH
Measures (2019), and we include what we need to be self-contained. Finally, we
state a conjectural combinatorial "formula" for the ideal of a generalized
orbital variety in terms of its tableau.
