Small resolutions of closures of K-orbits in flag varieties

Abstract

We construct explicit proper morphisms \(\mu\,\colon Z\to Y\), where \(Y\) is the closure of a \(K\)-orbit in a flag variety. Of particular interest is when \(\mu\) is a resolution of singularities and when it is a small resolution. We apply our construction to the group \(Sp(2n,\bR)\) and construct a resolution uniformly for every \(K\)-orbit closure in an isotropic grassmannian flag variety. This provides a family of small resolutions, and we change the construction to describe more families of small resolutions. We also apply our construction to the group \(U(p,q)\) and determine that any of our morphisms which are generically finite, are in fact birational for this group. This enables us to compute many small resolutions, and a simple family of small resolutions is described in terms of combinatorics of clans.

The concept of inducing a small resolution to larger dimensions is introduced. This shows that small resolutions propagate and highlights the importance of determining small resolutions in low rank groups. We apply this to the groups \(Sp(2n,\bR)\) and \(U(p,q)\) to obtain many small resolutions.

A repeated obstacle in applications is determining information about the fiber of \(\mu\). We provide a fiber dimension formula for a large class of our resolutions, which we call Barbasch-Evens type. When \(Z\) is constructed from a \(K\)-orbit closure and a single Schubert variety, then we describe fibers of \(\mu\) isomorphically -- enabling us to find more small resolutions and compute Kazhdan-Lusztig-Vogan polynomials of a family of closures of \(K\)-orbits.