On reductive subgroups of reductive groups having invariants in almost all representations

Abstract

Let $G$ and $\tilde G$ be connected complex reductive Lie groups, $G$ semisimple. Let $\Lambda^+$ be the monoid of dominant weights for a positive root system $\Delta^+$, and let $l(w)$ be the length of a Weyl group element $w$. Let $V_\lambda$ denote an irreducible $G$-module of highest weight $\lambda\in\Lambda^+$. For any closed embedding $\iota:\tilde G\subset G$, we consider Property (A): $\quad\forall\lambda\in\Lambda^+,\exists q\in\mathbb{N}$ such that $V_{q\lambda}^{\tilde G}\ne0$. A necessary condition for (A) is for $G$ to have no simple factors to which $G$ projects surjectively. We show that this condition is sufficient if $\tilde G$ is of type ${\bf A}_1$ or ${\bf E}_8$. We define and study an integral invariant of a root system, $\ell_G=\min\{\ell^\lambda:\lambda\in\Lambda^+\setminus\{0\}\}$, where $\ell^\lambda=\min\{l(w):w\lambda\notin{\rm Cone}(\Delta^+)\}$. We derive the following sufficient condition for (A), independent of $\iota$: $$ \ell_G - \#\tilde\Delta^+ > 0 \;\Longrightarrow\; (A). $$ We compute $\ell_G$ and related data for all simple $G$, except ${\bf E}_8$, where we obtain lower and upper bounds. We consider a stronger property (A-$k$) defined in terms of Geometric Invariant Theory, related to extreme values of codimensions of unstable loci, and derive a sufficient condition in the form $\ell_G - \#\tilde\Delta^+ > k$. The invariant $\ell_G$ proves too week to handle $G=SL_n$ and we employ a companion $\ell_G^{\rm sd}$ to infer (A-$k$) for a larger class of subgroups. We derive corollaries on Mori-theoretic properties of GIT-quotients.