Comparing two perfect bases

Abstract

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.