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

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.