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 \[ X_{d\lambda}=G\cdot[v_{d\lambda}] \subset \mathbb P V_{d\lambda} \] 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 $T^pX_{d\lambda}$. We prove that, if $d>4p$, its quadratic hull coincides with the osculating variety itself: \[ \mathcal Q(T^pX_{d\lambda})=T^pX_{d\lambda}. \] 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.