In recent work ([9], [10]), Kostant and Wallach construct an action of a simply connected Lie group A ≃ C ( n 2 ) on gl(n) using a completely integrable system derived from the Poisson analogue of the Gelfand-Zeitlin subalgebra of the enveloping algebra. In [9], the authors show that A-orbits of dimension n 2 form Lagrangian submanifolds of regular adjoint orbits in gl(n). They describe the orbit structure of A on a certain Zariski open subset of regular semisimple elements. In this paper, we describe all A-orbits of dimension n 2 and thus all polarizations of regular adjoint orbits obtained using Gelfand-Zeitlin theory. c . The construction in section 4 gives a bijection between A-orbits in gl(n) sreg c and orbits of a product of connected, commutative algebraic groups acting freely on a fairly simple variety, but it does not enumerate the A-orbits in gl(n) sreg c . In section 5, we use the construction developed in section 4 and combinatorial data of the fibre gl(n) sreg c to give explicit descriptions of the A-orbits in gl(n) sreg c. The main result is Theorem 5.11, which contrasts substantially with the generic case described in Theorem 1.1.be such that there are 0 ≤ j i ≤ i roots in common between the monic polynomials p c i (t) and p c i+1 (t). Then the number of A-orbits in gl(n) sreg c is exactly 2, let Z i denote the centralizer of the Jordan form of x i in gl(i). The orbits of A on gl(n) sreg c are the orbits of a free algebraic action of the complex, commutative, connected algebraic groupRemark 1.3. After the results of this paper were established, a very interesting paper by Roger Bielawski and Victor Pidstrygach appeared in [1] proving similar results. The arguments are completely different, and the proofs were formed independently. In [1], the authors define an action of A on the space of rational maps of fixed degree from the Riemann sphere into the flag manifold for GL(n + 1) and use symplectic reduction to obtain results about the strongly regular set. They also show that there are 2, c as in Theorem 1.2. Our work differs from that of [1] in that we explicitly list the A-orbits in gl(n) sreg c and obtain an algebraic action ofwhose orbits are the same as those of A. In spite of the relation between these papers, we feel that our paper provides a different and more precise perspective on the problem and deserves a place in the literature. c are A-stable. Moreover, Theorem 3.12 in [9] implies
Abstract. We generalize a result of Kostant and Wallach concerning the algebraic integrability of the Gelfand-Zeitlin vector fields to the full set of strongly regular elements in gl(n, C). We use decomposition classes to stratify the strongly regular set by subvarieties X D . We construct anétale coverĝ D of X D and show that X D andĝ D are smooth and irreducible. We then use Poisson geometry to lift the Gelfand-Zeitlin vector fields on X D to Hamiltonian vector fields onĝ D and integrate these vector fields to an action of a connected, commutative algebraic group.
In two 2006 papers, Kostant and Wallach constructed a complexified Gelfand-Zeitlin integrable system for the Lie algebra gl(n + 1, C) and introduced the strongly regular elements, which are the points where the Gelfand-Zeitlin flow is Lagrangian. Later Colarusso studied the nilfiber, which consists of strongly regular elements such that each i × i submatrix in the upper left corner is nilpotent. In this paper, we prove that every Borel subalgebra contains strongly regular elements and determine the Borel subalgebras containing elements of the nilfiber by using the theory of K i = G L(i − 1, C) × G L(1, C)-orbits on the flag variety for gl(i, C) for 2 ≤ i ≤ n + 1. As a consequence, we obtain a more precise description of the nilfiber. The K i -orbits contributing to the nilfiber are closely related to holomorphic and anti-holomorphic discrete series for the real Lie groups U (i, 1), with i ≤ n.
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