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Standard modules for rational Cherednik category O
Definition
Fix a finite-dimensional representation of . Regard as a module over the subalgebra of by letting act as given and every act as zero. The standard module associated to is By the PBW theorem The PBW theorem for rational Cherednik algebras, multiplication identifies its underlying vector space with . This item defines the induced module and vector-space description; its membership in is proved in Standard modules lie in rational Cherednik category O.
Remarks
No assumption that is irreducible is required for the induction. The irreducible standard labels are the finite-dimensional irreducible -modules.
Depends on
Used by
- Category O is not semisimple at the rank-one parameter c=1/2 Counterexample
- The polynomial representation of a rational Cherednik algebra Definition
- Cherednik costandards have orthogonality and the standard character Lemma
- Every rational Cherednik category-O object has finite length Lemma
- The standard-module KZ connection has the reflection residue Lemma
- Cherednik category O has a standard-filtered projective generator Theorem
- KZ monodromy of standard modules satisfies the reflection Hecke relations Theorem
- Standard modules lie in rational Cherednik category O Theorem
Dependency tree · two levels
5 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 and Xiaoguang Ma, Lecture Notes on Cherednik Algebras (standard reference, not scraped)
- Victor Ginzburg, Nicolas Guay, Eric Opdam, and Raphael Rouquier, On the category O for rational Cherednik algebras (standard reference, not scraped)