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The strong linkage principle for Verma modules
Statement
Assume the Axiom of Choice.
If , then .
Facts & Assumptions
Given: The Axiom of Choice (The Axiom of Choice), strong linkage The strong linkage order on weights, finite multiplicities under that Choice premise Verma composition multiplicities are finite, the first filtration term The first Jantzen filtration term is the maximal Verma submodule, and the Jantzen sum formula The Jantzen sum formula for a Verma module.
Proof
Induct on the height of . Height zero gives and the empty linkage chain.
For positive height, the factor is in the maximal submodule . The Jantzen sum and finite multiplicities place it in some with .
The new difference has smaller height, so induction gives ; appending the indicated positive-integral reflection gives .
Depends on
Used by
- A Verma embedding implies strong linkage Lemma
- Dominant integral dot translates embed canonically in the Verma module Lemma
- Jordan-Holder factors of Verma modules dominate the head (BGG 8.12) Lemma
- Separated integral cosets give typed parabolic category equivalences Lemma
- Simple extensions cannot cross linkage classes Lemma
- Wall-crossing compositions produce the projective with the top standard label Lemma
Dependency tree · two levels
20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Pavel Etingof, Representations of Lie Groups, Theorem 20.13 (standard reference, not scraped)