A restriction theorem for modules having a spherical submodule

Abstract:

We introduce the following notion: a finite dimensional representation $V$ of a complex reductive algebraic group $G$ is called spherical of rank one if the generic stabilizer $M$ is reductive, the pair $(G,M)$ is spherical and $\dim \;{V^M}= 1$. Let $U$ be another finite dimensional representation of $G$; we denote by $S’(U)\;(S’{(U)^G})$ the ring of polynomial functions on $U$ (the ring of $G$-invariant polynomial functions on $U$). We characterize the image of $S’{(U \oplus V)^G}$ under the restriction map into $S’ (U \oplus {V^M})$ as the $W= {N_G}(M)/M$ invariants in the Rees ring associated to an ascending filtration of $S’{(U)^M}$. Furthermore, under some additional hypothesis, we give an isomorphism between the graded ring associated to that filtration and $S’{(U)^P}$, where $P$ is the stabilizer of an unstable point whose $G$-orbit has maximal dimension.