DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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The Harish-Chandra projection

Definition

Fix a triangular decomposition $g = n^{-} \oplus h \oplus n^{+}$ from Triangular decomposition from a chosen positive root system. By the PBW basis from PBW gives an ordered monomial basis for the enveloping algebra, multiplication gives a vector-space decomposition

$U (g) = U (h) \oplus (n^{-} U (g) + U (g) n^{+}) .$

The projection onto the first summand is the Harish-Chandra projection

$pr ⁣ : U (g)_{0} \to U (h) = S (h),$

where $U (g)_{0}$ denotes the zero-weight subspace for the adjoint action of $h$ .

Depends on

Used by

Dependency tree · two levels

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Sources

Pavel Etingof, Representations of Lie Groups (standard reference, not scraped)
Lin Chen, Geometric Representation Theory I, Lecture 4 (standard reference, not scraped)