Restriction theorems for homogeneous bundles

Abstract

We prove that for an irreducible representation $\tau:GL(n)\to GL(W)$, the associated homogeneous ${\bf P}_k^n$-vector bundle $W_{\tau}$ is strongly semistable when restricted to any smooth quadric or to any smooth cubic in ${\bf P}_k^n$, where $k$ is an algebraically closed field of characteristic $\neq 2,3$ respectively. In particular $W_{\tau}$ is semistable when restricted to general hypersurfaces of degree $\geq 2$ and is strongly semistable when restricted to the $k$-generic hypersurface of degree $\geq 2$.