2021/05/06 by Dmitri I. Panyushev, Panyushev, Dmitri I.
Mathematics · #14L30 #14N07 #17B08 #17B70 #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT #msc:14L30 #msc:14N07 #msc:17B08 #msc:17B70
paper · pdf · doi:10.48550/arxiv.2105.02709
32 pages
arxiv created 2021/05/06 · arxiv updated 2021/05/07
Let G be a simple algebraic group with \mathfrak g=Lie G and \mathcal O\sf min⊂\mathfrak g the minimal nilpotent orbit. For a \mathbb Z2-grading \mathfrak g=\mathfrak g0⊕\mathfrak g1, let G0 be a connected subgroup of G with Lie G0=\mathfrak g0. We study the G0-equivariant projections φ:\mathcal O\sf min→ \mathfrak g0 and ψ:\mathcal O\sf min→\mathfrak g1. It is shown that the properties of φ(\mathcal O\sf min) and ψ(\mathcal O\sf min) essentially depend on whether the intersection \mathcal O\sf min∩\mathfrak g1 is empty or not. If \mathcal O\sf min∩\mathfrak g1≠\varnothing, then both φ(\mathcal O\sf min) and ψ(\mathcal O\sf min) contain a 1-parameter family of closed G0-orbits, while if \mathcal O\sf min∩\mathfrak g1=\varnothing, then both are G0-prehomogeneous. We prove that G⋅φ(\mathcal O\sf min)=G⋅ψ(\mathcal O\sf min). Moreover, if \mathcal O\sf min∩\mathfrak g1≠\varnothing, then this common variety is the affine cone over the secant variety of \mathbb P(\mathcal O\sf min)⊂\mathbb P(\mathfrak g). As a digression, we obtain some invariant-theoretic results on the affine cone over the secant variety of the minimal orbit in an arbitrary simple G-module. In conclusion, we discuss more general projections that are related to either arbitrary reductive subalgebras of \mathfrak g in place of \mathfrak g0 or spherical nilpotent G-orbits in place of \mathcal O\sf min.