DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-09-06

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The one-dimensional Borel module of weight lambda

Definition

Fix the Borel subalgebra $b = h \oplus n^{+}$ from Triangular decomposition from a chosen positive root system. For $λ \in h^{*}$ , let $C_{λ} = C c_{λ}$ be the one-dimensional $b$ -module defined by

$h c_{λ} = λ (h) c_{λ} (h \in h), x c_{λ} = 0 (x \in n^{+}) .$

This is well defined: $[h, n^{+}] \subseteq n^{+}$ , and $[n^{+}, n^{+}] \subseteq n^{+}$ , so both sides of the representation identity vanish on $c_{λ}$ whenever one input lies in $n^{+}$ . The weight is unshifted.

Depends on

Triangular decomposition from a chosen positive root system

Used by

Verma modules Definition

Dependency tree · two levels

4 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, Lie Groups and Lie Algebras I & II, Remark 25.8 (standard reference, not scraped)