Eigenvalue coincidences and K-orbits, I

Abstract

We study the variety g(l) consisting of matrices x∈gl(n,C) such that x and its n−1 by n−1 cutoff xn−1 share exactly l eigenvalues, counted with multiplicity. We determine the irreducible components of g(l) by using the orbits of GL(n−1,C) on the flag variety B of gl(n,C). More precisely, let b∈B be a Borel subalgebra such that the orbit GL(n−1,C)⋅b in B has codimension l. Then we show that the set Yb:={Ad(g)(x):x∈b∩g(l),g∈GL(n−1,C)} is an irreducible component of g(l), and every irreducible component of g(l) is of the form Yb, where b lies in a GL(n−1,C)-orbit of codimension l. An important ingredient in our proof is the flatness of a variant of a morphism considered by Kostant and Wallach, and we prove this flatness assertion using an analogue of the Steinberg variety.