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Generic stabilizers for simple algebraic groups acting on orthogonal and symplectic Grassmannians

Aluna Rizzoli

TL;DR

This work classifies generic and semi-generic stabilizers for the action of simple algebraic groups on orthogonal and symplectic Grassmannians of self-dual irreducible modules under the regime \(\dim G \geq \dim \mathcal{S}_k(V)\). It develops a robust framework using bilinear forms, Clifford theory, generic stabilizers, and spin modules to reduce to ts-small quadruples, then treats finitely many cases with parabolic stabilizers and several remaining cases via a combination of theoretical and Magma-assisted computations. The main contributions include a complete list of (semi-)generic ts-stabilizers for ts-small quadruples, a precise criterion for the existence of dense orbits on orthogonal/symplectic Grassmannians, and a classification of pairs of maximal connected subgroups with dense double cosets in classical groups. These results illuminate the structure of action orbits and have implications for density questions in prehomogeneous vector spaces and for the double coset density problem in classical groups, with explicit handling of the tricky exceptional cases via computational methods. The findings advance understanding of when dense orbits and dense double cosets arise, enabling broader applications in representation theory and algebraic group actions.

Abstract

We consider faithful actions of simple algebraic groups on self-dual irreducible modules, and on the associated varieties of totally singular subspaces, under the assumption that the dimension of the group is at least as large as the dimension of the variety. We prove that in all but a finite list of cases, there is a dense open subset where the stabilizer of any point is conjugate to a fixed subgroup, called the generic stabilizer. We use these results to determine whether there exists a dense orbit. This in turn lets us complete the answer to the problem of determining all pairs of maximal connected subgroups of a classical group with a dense double coset.

Generic stabilizers for simple algebraic groups acting on orthogonal and symplectic Grassmannians

TL;DR

This work classifies generic and semi-generic stabilizers for the action of simple algebraic groups on orthogonal and symplectic Grassmannians of self-dual irreducible modules under the regime \(\dim G \geq \dim \mathcal{S}_k(V)\). It develops a robust framework using bilinear forms, Clifford theory, generic stabilizers, and spin modules to reduce to ts-small quadruples, then treats finitely many cases with parabolic stabilizers and several remaining cases via a combination of theoretical and Magma-assisted computations. The main contributions include a complete list of (semi-)generic ts-stabilizers for ts-small quadruples, a precise criterion for the existence of dense orbits on orthogonal/symplectic Grassmannians, and a classification of pairs of maximal connected subgroups with dense double cosets in classical groups. These results illuminate the structure of action orbits and have implications for density questions in prehomogeneous vector spaces and for the double coset density problem in classical groups, with explicit handling of the tricky exceptional cases via computational methods. The findings advance understanding of when dense orbits and dense double cosets arise, enabling broader applications in representation theory and algebraic group actions.

Abstract

We consider faithful actions of simple algebraic groups on self-dual irreducible modules, and on the associated varieties of totally singular subspaces, under the assumption that the dimension of the group is at least as large as the dimension of the variety. We prove that in all but a finite list of cases, there is a dense open subset where the stabilizer of any point is conjugate to a fixed subgroup, called the generic stabilizer. We use these results to determine whether there exists a dense orbit. This in turn lets us complete the answer to the problem of determining all pairs of maximal connected subgroups of a classical group with a dense double coset.
Paper Structure (15 sections, 83 theorems, 104 equations)

This paper contains 15 sections, 83 theorems, 104 equations.

Table of Contents

  1. Introduction and statement of results
  2. Preliminary results
  3. Bilinear forms
  4. Clifford theory
  5. Generic stabilizers
  6. Spin modules
  7. List of $ts$-small quadruples
  8. Quadruples with finitely many orbits on $\mathcal{G}_k(V)$
  9. Infinite families of quadruples
  10. Remaining quadruples
  11. The case $(C_2, 2\lambda_1,p,5)$
  12. The case $(C_3, \lambda_2,p,7)$
  13. Proof of Theorem \ref{['maximal semisimple theorem']}
  14. Proof of Theorem \ref{['theorem double coset density']}
  15. Magma code

Key Result

Theorem 1

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p$, and $V$ a self-dual non-trivial irreducible $G$-module of dimension $d$ and highest weight $\lambda$. For $1\leq k\leq \frac{d}{2}$ such that $\dim G\geq \dim \mathcal{S}_k(V)$, if the action of $G$ on $\ma In the first three cases the action of $G$ on $\mathcal{S}_k(V)$ has no generic stabilizer, but doe

Theorems & Definitions (158)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Theorem 4
  • Theorem 5
  • Remark 2
  • Remark 3
  • Conjecture 1
  • Theorem 2.1
  • ...and 148 more