Eigenvalue coincidences and K-orbits, I

We study the variety g(l)g(l) consisting of matrices x∈gl(n,C)x∈gl(n,C) such that x   and its n−1n−1 by n−1n−1 cutoff xn−1xn−1 share exactly l   eigenvalues, counted with multiplicity. We determine the irreducible components of g(l)g(l) by using the orbits of GL(n−1,C)GL(n−1,C) on the flag variety BB of gl(n,C)gl(n,C). More precisely, let b∈Bb∈B be a Borel subalgebra such that the orbit GL(n−1,C)⋅bGL(n−1,C)⋅b in BB has codimension l  . Then we show that the set Yb:={Ad(g)(x):x∈b∩g(l),g∈GL(n−1,C)}Yb:={Ad(g)(x):x∈b∩g(l),g∈GL(n−1,C)} is an irreducible component of g(l)g(l), and every irreducible component of g(l)g(l) is of the form YbYb, where bb lies in a GL(n−1,C)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.