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

TL;DR

This paper investigates conditions under which invariants in almost all representations exist for certain subgroup embeddings of complex reductive Lie groups, introducing new invariants and criteria to understand these properties.

Contribution

It introduces an integral invariant of root systems and provides sufficient conditions for the existence of invariants in representations, extending understanding of reductive subgroup embeddings.

Findings

Established a necessary and sufficient condition for property (A) in specific cases.

Computed the invariant ll_G for all simple groups except E8.

Derived corollaries related to Mori-theoretic properties of GIT-quotients.

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.