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Sam Evens

  1. The Gelfand-Zeitlin integrable system and K-orbits on the flag variety.

    Втр, 11/15/2011 - 15:01 — SomNambul
    Authors: Mark Colarusso, Sam Evens
    Subjects: Representation Theory
    Abstract

    In this expository paper, we provide an overview of the Gelfand-Zeiltin
    integrable system on the Lie algebra of $n\times n$ complex matrices
    $\fgl(n,\C)$ introduced 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 Gelfand-Zeitlin flow is Lagrangian. We use the theory of
    $K_{n}=GL(n-1,\C)\times GL(1,\C)$-orbits on the flag variety $\mathcal{B}_{n}$
    of $GL(n,\C)$ to describe the strongly regular elements in the nilfiber of the
    moment map of the system.

  2. On some invariants of orbits in the flag variety under a symmetric subgroup.

    Пт, 04/15/2011 - 15:01 — SomNambul
    Authors: Sam Evens, Jiang-Hua Lu
    Subjects: Representation Theory
    Abstract

    Let $G$ be a connected reductive algebraic group over an algebraically closed
    field ${\bf k}$ of characteristic not equal to 2, let $\B$ be the variety of
    all Borel subgroups of $G$, and let $K$ be a symmetric subgroup of $G$. Fixing
    a closed $K$-orbit in $\B$, we associate to every $K$-orbit on $\B$ some
    subsets of the Weyl group of $G$, and we study them as invariants of the
    $K$-orbits. When ${\bf k} = {\mathbb C}$, these invariants are used to
    determine when an orbit of a real form of $G$ and an orbit of a Borel subgroup
    of $G$ have non-empty intersection in $\B$.

  3. On Algebraic Integrability of Gelfand-Zeitlin fields.

    Чт, 08/27/2009 - 13:23 — SomNambul
    Authors: Mark Colarusso, Sam Evens
    Subjects: Symplectic Geometry
    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 $gl(n,\mathbb{C})$. We use decomposition classes to
    stratify the strongly regular set by subvarieties $X_{D}$. We construct an
    \'{e}tale cover $\hat{\mathfrak{g}}$ of $X_{D}$ and show that $X_{D}$ and
    $\hat{\mathfrak{g}}$ are smooth and irreducible.

  4. On Algebraic Integrability of Gelfand-Zeitlin fields.

    Чт, 08/27/2009 - 13:23 — SomNambul
    Authors: Mark Colarusso, Sam Evens
    Subjects: Symplectic Geometry
    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 $gl(n,\mathbb{C})$. We use decomposition classes to
    stratify the strongly regular set by subvarieties $X_{D}$. We construct an
    \'{e}tale cover $\hat{\mathfrak{g}}$ of $X_{D}$ and show that $X_{D}$ and
    $\hat{\mathfrak{g}}$ are smooth and irreducible.

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