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

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


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.


Poisson Bracket Spectral Sequence Weyl Algebra Nilpotent Orbit Coordinate Ring 
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© 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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