Irreducibility of quadratic hulls of osculating varieties of closed orbits

Abstract

Let $G$ be a connected complex semisimple algebraic group, let $V_{\lambda }$ be an irreducible $G$-module of highest weight $\lambda$, and let \[ X_{dλ}=G\cdot[v_{dλ}] \subset \mathbb P V_{dλ} \] be the closed orbit of a highest weight vector. We study the irreducibility of the zero locus of the quadratic equations containing the $p$-osculating variety $T^p{X}_{d\lambda }$. We prove that, if $d>4p$, its quadratic hull coincides with the osculating variety itself: \[ \mathcal Q(T^pX_{dλ})=T^pX_{dλ}. \] In particular, the quadratic hull is irreducible. Thus, for every fixed $p$, sufficiently high Cartan powers of a closed highest weight orbit have $p$-osculating varieties determined set-theoretically by their quadratic equations.

arXiv commentMajor revision. The main irreducibility theorem in the previous version is false in the stated generality. The error occurs in Step 4 of the proof of Theorem 6. The present version replaces the incorrect general statement by a valid result for quadratic hulls of osculating varieties of closed orbits under suitable hypotheses