Closed Orbits and Descents for Enhanced Standard Representations of Classical Groups

2025/07/29 by Liang, Chen
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2507.21911

Abstract

Let G=GL_n(\mathbbF), O_n(\mathbbF), or Sp_2n(\mathbbF) be one of the classical groups over an algebraically closed field \mathbbF of characteristic 0, let \breveG be the MVW-extension of G, and let \mathfrakg be the Lie algebra of G. In this paper, we classify the closed orbits in the enhanced standard representation \mathfrakg× E of G, where E is the natural representation if G=O_n(\mathbbF) or Sp_2n(\mathbbF), and is the direct sum of the natural representation and its dual if G=GL_n(\mathbbF). Additionally, for every closed G-orbit in \mathfrakg× E, we prove that it is \breveG-stable, and determine explicitly the corresponding stabilizer group as well as the action on the normal space.

Citations

• Relative Langlands Duality
• Le transfert singulier pour la formule des traces de Jacquet-Rallis
• Enhanced orbit embedding
• Notes on MVW-extensions
• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
• Multiplicity one Conjectures

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• Arithmetic invariant theory