Minimal Nilpotent Orbits and Toric Varieties

TL;DR

This paper studies a specific nilpotent orbit intersection in $rak{sl}_{n+1}(C)$, showing it degenerates to a toric variety, computing its Hilbert series, and proving it is reduced and Gorenstein, with implications for orbital varieties.

Contribution

It introduces a flat degeneration of a nilpotent orbit intersection to a toric variety and computes its Hilbert series, establishing new geometric and algebraic properties.

Findings

The intersection has a flat degeneration to a toric variety.

The coordinate ring is reduced, Gorenstein, and Cohen-Macaulay.

Explicit Hilbert series are computed for the coordinate ring.

Abstract

Let $\overline{\mathcal{O}}_\textrm{min} \cap (\mathfrak n^+ \oplus \mathfrak n^-)$ be the collection of elements of $\mathfrak{sl}_{n+1}(\mathbb C)$ with rank less than or equal to $1$ and with all diagonal entries equal to zero. We show that the coordinate ring $\mathbb C[\overline{\mathcal{O}}_\textrm{min} \cap (\mathfrak n^+ \oplus \mathfrak n^-)]$ of the scheme-theoretic intersection $\overline{\mathcal{O}}_\textrm{min} \cap (\mathfrak n^+ \oplus \mathfrak n^-)$ has a flat degeneration to the ring of $(\mathbb C^{\times})^n$-equivariant cohomology of the projective toric variety associated with the fan of compatible subsets of almost positive roots of type $C_n$. Then we compute the Hilbert series of $\mathbb C[\overline{\mathcal{O}}_\textrm{min} \cap (\mathfrak n^+ \oplus \mathfrak n^-)]$ and prove that $\overline{\mathcal{O}}_\textrm{min} \cap (\mathfrak n^+ \oplus \mathfrak n^-)$ is reduced and Gorenstein. Moreover, our proof method allows us to prove that the scheme-theoretic intersection $\overline{\mathcal{O}}_\textrm{min} \cap \mathfrak n^+$, of which the irreducible components are known as the ``orbital varieties'', is reduced and Cohen-Macaulay.