• Corpus ID: 239049440

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

@inproceedings{Tsanov2021OnRS,
  title={On reductive subgroups of reductive groups having invariants in almost all representations},
  author={Valdemar V. Tsanov and Yana Staneva},
  year={2021}
}

• V. Tsanov, Yana Staneva
• Published 21 October 2021
• Mathematics

Let G and G̃ be connected complex reductive Lie groups, G semisimple. Let Λ+ be the monoid of dominant weights for a positive root system ∆+, and let l(w) be the length of a Weyl group element w. Let Vλ denote an irreducible G-module of highest weight λ ∈ Λ+. For any closed embedding ι : G̃ ⊂ G, we consider Property (A): ∀λ ∈ Λ+,∃q ∈ N such that V G̃ qλ 6= 0. 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 G̃ is of type A1 or E8. We define and study an integral invariant of a root system, lG = min{l λ : λ ∈ Λ+\{0}}, where lλ = min{l(w) : wλ / ∈ Cone(∆+)}. We derive the following sufficient condition for (A), independent of ι: lG −#∆̃ + > 0 =⇒ (A). We compute lG and related data for all simple G, except E8, 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 lG−#∆̃ + > k. The invariant lG proves too week to handle G = SLn and we employ a companion l sd G to infer (A-k) for a larger class of subgroups. We derive corollaries on Mori-theoretic properties of GIT-quotients.