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Nilpotent orbits of height 2 and involutions in the affine Weyl group

Jacopo Gandini, Pierluigi Moseneder Frajria, Paolo Papi

Abstract

Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B be a Borel subgroup of G. Then B acts with finitely many orbits on the variety N_2 of the nilpotent elements in g whose height is at most 2. We provide a parametrization of the B-orbits in N_2 in terms of subsets of pairwise orthogonal roots, and we provide a complete description of the inclusion order among the B-orbit closures in terms of the Bruhat order on certain involutions in the affine Weyl group of g.

Nilpotent orbits of height 2 and involutions in the affine Weyl group

Abstract

Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B be a Borel subgroup of G. Then B acts with finitely many orbits on the variety N_2 of the nilpotent elements in g whose height is at most 2. We provide a parametrization of the B-orbits in N_2 in terms of subsets of pairwise orthogonal roots, and we provide a complete description of the inclusion order among the B-orbit closures in terms of the Bruhat order on certain involutions in the affine Weyl group of g.

Paper Structure

This paper contains 8 sections, 39 theorems, 95 equations, 1 table.

Table of Contents

  1. Notation and preliminaries.
  2. Nilpotent elements associated to strongly orthogonal sets of roots
  3. The closure of a nilpotent orbit of height 2 and its resolution
  4. Orbit decompositions of $X$, $\widetilde{X}$, and $\mathfrak a$
  5. Canonical orthogonal subsets and the corresponding characteristics
  6. The fibers of the resolution.
  7. The Bruhat order on $\widetilde{X}$
  8. The Bruhat order on $\mathcal{N}_2$.

Key Result

Theorem 1

Let $R,S \subset \Phi$ be strongly orthogonal with $\mathop{\mathrm{ht}}\nolimits(e_R) = \mathop{\mathrm{ht}}\nolimits(e_S) = 2$, then $Be_R \subset \overline{Be_S}$ if and only if $\sigma_{\widehat{R}} \leqslant \sigma_{\widehat{S}}$. Moreover, we have

Theorems & Definitions (77)

  • Theorem 1: see Corollary \ref{['cor:dim-formula']} and Theorem \ref{['teo:bruhat3']}
  • Theorem 2: see Theorem \ref{['teo:bruhat1']}
  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Proposition 2.1: see GMP
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • ...and 67 more