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Dixmier Algebras for Classical Complex Nilpotent Orbits via Kraft-Procesi Models I

  • Ranee Brylinski
Chapter
Part of the Progress in Mathematics book series (PM, volume 213)

Abstract

We attach a Dixmier algebra \(\mathcal{B}\) to the closure of \(\bar{\mathcal{O}}\) any nilpotent orbit of G where G is GL(nℂ), O(nℂ) or Sp(2n ℂ). This algebra \(\mathcal{B}\) is a noncommutative analog of the coordinate ring \(\mathcal{R} of \bar{\mathcal{O}}\) in the sense that \(\mathcal{B}\) has a G-invariant algebra filtration and \(gr\mathcal{B} = \mathcal{R}\).

We obtain \(\mathcal{B}\) by making a noncommutative analog of the Kraft-Procesi construction which modeled \(\bar{\mathcal{O}}\) as the algebraic symplectic reduction of a finite-dimensional symplectic vector space L. Indeed \(\mathcal{B}\) is a subquotient of the Weyl algebra for L.

\(\mathcal{B}\) identifies with the quotient of \(\mathcal{U}{\text{(}}\mathfrak{g})\) by a two-sided ideal J, where \(\mathfrak{g} = Lie (G)\). Then gr J is the ideal \(\Im (\bar{\mathcal{O}})\) in \(S(\mathfrak{g})\) of functions vanishing on \(\bar{\mathcal{O}}\). In every case where \(\mathcal{O}\) is connected, J is a completely prime primitive ideal.

Keywords

Poisson Bracket Spectral Sequence Weyl Algebra Nilpotent Orbit Coordinate Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Ranee Brylinski
    • 1
  1. 1.Department of MathematicsPenn State UniversityUniversity ParkUSA

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