2013/03/26 by Mark Colarusso, Colarusso, Mark, Sam Evens +1
Mathematics · #14L30 #14M15 #20G20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT #msc:14L30 #msc:14M15 #msc:20G20
paper · pdf · doi:10.48550/arxiv.1303.6661
17 pages
arxiv created 2013/04/24 · arxiv updated 2013/04/26
We study the variety \mathfrakg(l) consisting of matrices x ∈ \mathfrakgl(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 \mathfrakg(l) by using the orbits of GL(n-1,\C) on the flag variety \Bn of \mathfrakgl(n,\C). More precisely, let \mathfrakb ∈ \Bn be a Borel subalgebra such that the orbit GL(n-1,\C)⋅ \mathfrakb in \Bn has codimension l. Then we show that the set Y\fb:= \\Ad(g)(x): x∈ \mathfrakb ∩ \mathfrakg(l), g∈ GL(n-1,\C)\ is an irreducible component of \mathfrakg(l), and every irreducible component of of \mathfrakg(l) is of the form Y_\mathfrakb, where \mathfrakb 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 ideas from symplectic geometry.