2014/10/15 by Mark Colarusso, Colarusso, Mark, Sam Evens +1 · 1 citation
Mathematics · #14L30 #14M15 #20G20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1410.3901
openalex publication_date 2014/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In recent work, we related the structure of subvarieties of n× n complex matrices defined by eigenvalue coincidences to GL(n-1,ℂ)-orbits on the flag variety of \mathfrakgl(n,ℂ). In the first part of this paper, we extend these results to the complex orthogonal Lie algebra \mathfrakg=\mathfrakso(n,ℂ). In the second part of the paper, we use these results to study the geometry and invariant theory of the K-action on \mathfrakg, in the cases where (\mathfrakg, K) is (\mathfrakgl(n,ℂ), GL(n-1,ℂ)) or (\mathfrakso(n,ℂ), SO(n-1,ℂ)). We study the geometric quotient \mathfrakg→ \mathfrakg//K and describe the closed K-orbits on \mathfrakg and the structure of the zero fibre. We also prove that for x∈ \mathfrakg, the K-orbit Ad(K)⋅ x has maximal dimension if and only if the algebraically independent generators of the invariant ring ℂ[\mathfrakg]K are linearly independent at x, which extends a theorem of Kostant. We give applications of our results to the Gelfand-Zeitlin system.