The Gelfand–Zeitlin integrable system and K-orbits on the flag variety Mark Colarusso and Sam Evens∗ To Nolan Wallach, on the occasion of his 70th birthday, with gratitude and admiration Abstract In this paper, we provide an overview of the Gelfand–Zeitlin in- tegrable system on the Lie algebra of n n complex matrices gl(n, C) in- troduced by Kostant and Wallach in 2006.× We discuss results concerning the geometry of the set of strongly regular elements, which consists of the points where the Gelfand–Zeitlin flow is Lagrangian. We use the theory of Kn = GL(n 1, C) GL(1, C)-orbits on the flag variety n of GL(n, C) to describe the− strongly× regular elements in the nilfiber of theB moment map of the system. We give an overview of the general theory of orbits of a sym- metric subgroup of a reductive algebraic group acting on its flag variety, and illustrate how the general theory can be applied to understand the specific example of Kn and GL(n, C). Key words: Flag variety • Symmetric subgroup • Nilpotent matrices • Integrable systems • Gelfand–Zeitlin theory Mathematics Subject Classification 2010: 20G20, 14M15, 14L30, 70H06, 17B08, 37J35 Mark Colarusso (B) Department Mathematical Sciences, University of Wisconsin–Milwaukee, Milwaukee, WI 53201-0413, e-mail:
[email protected] Sam Evens (B) Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, e-mail:
[email protected] ∗ The work by the second author was partially supported by NSA grants H98230-08-0023 and H98230-11-1-0151. 1 2 Mark Colarusso and Sam Evens 1 Introduction In a series of papers [24, 25], Kostant and Wallach study the action of an n(n−1) C 2 C abelian Lie group A ∼= on g = gl(n, ).