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• W44230476 abstract "Author(s): Colarusso, Mark | Abstract: The main results of this thesis describe and construct polarizations of regular adjoint orbits for certain classical groups. The thesis generalizes recent work of Bertram Kostant and Nolan Wallach (see [KW]). Kostant and Wallach construct polarizations of regular adjoint orbits for n x n complex matrices M(n). (Xin M(n) is said to be regular if it has a cyclic vector in C}n}.) They accomplish this by defining an n(n-1)/2 dimensional abelian complex Lie group A that acts on M(n) and stabilizes adjoint orbits. We study the A orbit structure on M(n) and generalize the construction to complex orthogonal Lie algebras mathfrak}so}(n). Since the A orbits of dimension n(n-1)/2 contained in a given regular adjoint orbit form the leaves of a polarization of an open submanifold of that orbit, we then obtain descriptions of these polarizations. For mathfrak}so}(n) we construct polarizations of certain regular semi-simple orbits. In the case of M(n), we study the action of A on matrices whose principal i x i submatrices in the top left hand corner (i x i cutoffs) are regular. We determine the A orbit structure of Zariski closed subvarieties of these matrices defined by each cutoff having a fixed characteristic polynomial. We relate the number of A orbits of maximal dimension n(n-1)/2 in such closed sets to the number of eigenvalues shared by adjacent cutoffs. The number of such A orbits always turns out to be a power of 2. We are also able to describe all of the A orbits of maximal dimension as orbits of Zariski connected algebraic groups acting on certain quasi-affine subvarieties of M(n). This work gives an explicit description of the leaves of polarizations of all regular adjoint orbits in M(n). In the case of the orthogonal Lie algebras, we construct a completely integrable system of commuting Hamiltonian vector fields on mathfrak}so}(n) using a classical analogue of the Gelfand-Zeitlin algebra of the universal enveloping algebra of mathfrak}so}(n) in the algebra of polynomials on mathfrak}so}(n). Integrating these vector fields gives rise to an action of C}d/2}, where d is the dimension of a regular adjoint orbit in an orthogonal Lie algebra. The action of C}d/2} stabilizes adjoint orbits. We then use the orbits of C}d/2} to construct polarizations of certain regular semi-simple adjoint orbits in mathfrak}so}(n). We realize the leaves of these polarizations as orbits of a Zariski connected algebraic group" @default.
• W44230476 created "2016-06-24" @default.
• W44230476 creator A5037564598 @default.
• W44230476 date "2007-01-01" @default.
• W44230476 modified "2023-09-27" @default.
• W44230476 title "The Gelfand -Zeitlin algebra and polarizations of regular adjoint orbits for classical groups" @default.
• W44230476 hasPublicationYear "2007" @default.
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