Weyl-invariant subspaces are (usually) not generic

Abstract

Let $V$ be a linear representation of a connected complex reductive group $G$. Given a choice of character $\hat{I}¸$ of $G$, Geometric Invariant Theory defines a locus $V_{\hat{I}}^{ss}¸(G)\subseteq V$ of semistable points. We give necessary, sufficient, and in some cases equivalent conditions for the existence of $\hat{I}¸$ such that a maximal torus $T$ of $G$ acts on $V_{\hat{I}}^{ss}¸(T)$ with finite stabilizers. In such cases, the stack quotient $[V_{\hat{I}}^{ss}¸(G)/G]$ is is known to be Deligne-Mumford. Our proof uses the combinatorial structure of the weights of irreducible representations of semisimple groups. As an application we generalize the Grassmannian flop example of Donovan-Segal.

26 pages