Back to Search
Start Over
Projections of the minimal nilpotent orbit in a simple Lie algebra and secant varieties.
- Source :
-
Mathematical Proceedings of the Cambridge Philosophical Society . Nov2023, Vol. 175 Issue 3, p595-624. 30p. - Publication Year :
- 2023
-
Abstract
- Let G be a simple algebraic group with ${\mathfrak g}={\textrm{Lie }} G$ and ${\mathcal O}_{\textsf{min}}\subset{\mathfrak g}$ the minimal nilpotent orbit. For a ${\mathbb Z}_2$ -grading ${\mathfrak g}={\mathfrak g}_0\oplus{\mathfrak g}_1$ , let $G_0$ be a connected subgroup of G with ${\textrm{Lie }} G_0={\mathfrak g}_0$. We study the $G_0$ -equivariant projections $\varphi\,:\,\overline{{\mathcal O}_{\textsf{min}}}\to {\mathfrak g}_0$ and $\psi:\overline{{\mathcal O}_{\textsf{min}}}\to{\mathfrak g}_1$. It is shown that the properties of $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ essentially depend on whether the intersection ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1$ is empty or not. If ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , then both $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ contain a 1-parameter family of closed $G_0$ -orbits, while if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1=\varnothing$ , then both are $G_0$ -prehomogeneous. We prove that $\overline{G{\cdot}\varphi({\mathcal O}_{\textsf{min}})}=\overline{G{\cdot}\psi({\mathcal O}_{\textsf{min}})}$. Moreover, if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , then this common variety is the affine cone over the secant variety of ${\mathbb P}({\mathcal O}_{\textsf{min}})\subset{\mathbb P}({\mathfrak g})$. 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 ${\mathfrak g}_0$ or spherical nilpotent G -orbits in place of ${\mathcal O}_{\textsf{min}}$. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 03050041
- Volume :
- 175
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Mathematical Proceedings of the Cambridge Philosophical Society
- Publication Type :
- Academic Journal
- Accession number :
- 172989929
- Full Text :
- https://doi.org/10.1017/S0305004123000348