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The Harish-Chandra projection
Definition
Fix a triangular decomposition 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
The projection onto the first summand is the Harish-Chandra projection
where denotes the zero-weight subspace for the adjoint action of .
Depends on
Used by
- Flag-variety twisted differential operators and their modules Definition
- A point-supported SL2 module gives the shifted Verma module Example
- Central actions on nilradical cohomology factor through the Harish–Chandra projection Lemma
- Central elements lie in the zero-weight subspace of U(g) Lemma
- Every central character of a semisimple enveloping algebra arises from a weight Lemma
- The central action on the deformed antidominant projective Lemma
- The Harish-Chandra projection computes the highest-weight scalar Lemma
- The Harish-Chandra map on the center is injective Proposition
- The Harish-Chandra projection is multiplicative on the center Proposition
- Generic irreducibility and the exceptional parameter lattice Theorem
- Global sections of twisted differential operators on the flag variety Theorem
- The Casselman–Osborne constraint on weights of nilradical cohomology Theorem
Dependency tree · two levels
6 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 (standard reference, not scraped)
- Lin Chen, Geometric Representation Theory I, Lecture 4 (standard reference, not scraped)