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The Bruhat order on abelian ideals of Borel subalgebras

Jacopo Gandini, Andrea Maffei, Pierluigi Moseneder Frajria, Paolo Papi

Abstract

Let G be a quasi simple algebraic group over an algebraically closed field k whose characteristic is not very bad for G, and let B be a Borel subgroup of G with Lie algebra b. Given a B-stable abelian subalgebra a of the nilradical of b, we parametrize the B-orbits in a and we describe their closure relations.

The Bruhat order on abelian ideals of Borel subalgebras

Abstract

Let G be a quasi simple algebraic group over an algebraically closed field k whose characteristic is not very bad for G, and let B be a Borel subgroup of G with Lie algebra b. Given a B-stable abelian subalgebra a of the nilradical of b, we parametrize the B-orbits in a and we describe their closure relations.

Paper Structure

This paper contains 9 sections, 20 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Abelian ideals of $\mathfrak b$ in positive characteristic
  4. Minuscule elements in the affine Weyl group
  5. Affine involutions associated to orthogonal sets of roots and admissible pairs.
  6. Admissible pairs and their associated involutions
  7. The extended action, and dimension formulas
  8. The Bruhat order of the $B_v$-orbits on $\widehat{\mathfrak a}_w$
  9. Further developments.

Key Result

Theorem 1

Let $\mathfrak a$ be $B$-stable abelian subalgebra of $\mathfrak u$ and let $S,S' \subset \Psi(\mathfrak a)$ be orthogonal subsets. Set $\widehat{S} = S-\delta$ and $\widehat{S}' = S'-\delta$, then Moreover,

Theorems & Definitions (44)

  • Theorem : see Corollary \ref{['cor:parametrizzazionedimensione']} and Theorem \ref{['teo:principale']}
  • Lemma 3.1
  • proof
  • Theorem 3.2: Pa4
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Remark 3.6
  • ...and 34 more