2024/10/13 by Dmitri I. Panyushev, Panyushev, Dmitri I.
Biochemistry, Genetics and Molecular Biology · Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometric and Algebraic Topology #Microtubule and mitosis dynamics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2410.09876
openalex publication_date 2024/10/13 · openalex created_date 2024/10/20 · openalex updated_date 2026/07/28
Let G be a simple algebraic group with \mathfrak g=Lie(G) and \mathcal O⊂\mathfrak g a nilpotent orbit. If H is a reductive subgroup of G with Lie(H)=\mathfrak h, then \mathfrak g=\mathfrak h⊕\mathfrak m, where \mathfrak m=\mathfrak h^⊥. We consider the natural projections ϕ: \mathcal O→\mathfrak h and ψ:\mathcal O→\mathfrak m, and two related properties of the pair (H,\mathcal O): (P1): \mathcal O∩\mathfrak m=0 and (P2): H has a dense orbit in \mathcal O. We show that (P1) implies (P2) for all \mathcal O and these properties are equivalent for \mathcal O=\mathcal Omin, the minimal nilpotent orbit. If (P1) holds, then ϕ is finite, and ϕ(\mathcal O) is the closure of a nilpotent H-orbit \mathcal O'. We prove that \mathcal O is contained in the closure of the G-orbit G⋅\mathcal O' and obtain the classification of pairs (H,\mathcal O) with property (P1). The orbit \mathcal O' is "shared" in the sense of Brylinski and Kostant. Using our classification, we detect an omission in the list of pairs (H,G) having a shared orbit that is given in "Nilpotent orbits, normality, and hamiltonian group actions", J.A.M.S., 7 (1994), 269--298. It is also proved that if (P1) holds for (H, \mathcal Omin), then both varieties ϕ(\mathcal Omin) and ψ(\mathcal Omin) generate the same closed subvariety of \mathfrak g.