[Paper Review] Induction and restriction functors for rational Cherednik algebras

Roman Bezrukavnikov, Pavel Etingof|arXiv (Cornell University)|Mar 25, 2008

Advanced Topics in Algebra8 citations

TL;DR

This paper introduces parabolic induction and restriction functors for rational Cherednik algebras, enabling a deeper structural analysis of their representation theory. It proves the Gordon-Stafford Morita equivalence without the prior restriction on the parameter c, classifies representations in category O without Sn-invariant vectors, and confirms a conjecture on their count, while also establishing simplicity of the spherical Cherednik algebra for −1 < c < 0.

ABSTRACT

We introduce parabolic induction and restriction functors for rational Cherednik algebras, and study their basic properties. Then we discuss applications of these functors to representation theory of rational Cherednik algebras. In particular, we prove the Gordon-Stafford theorem about Morita equivalence of the rational Cherednik algebra for type A and its spherical subalgebra, without the assumption that c is not a half-integer, which was required up to now. Also, we classify representations from category O over the rational Cherednik algebras of type A which do not contain an S_n-invariant vector, and confirm a conjecture of Okounkov and the first author on the number of such representations. In the second version we have added a result on the simplicity of the spherical Cherednik algebra of type A for -1<c<0, and a strengthened version of the main result of arXiv:math/0312474, as well as an appendix by the second author containing arXiv:0706.4308, on the reducibility of the polynomial representation of the trigonometric Cherednik algebra.

Motivation & Objective

• To develop parabolic induction and restriction functors for rational Cherednik algebras to enhance structural understanding of their representation theory.
• To remove the restriction that c ≠ half-integer in the Gordon-Stafford Morita equivalence theorem for type A rational Cherednik algebras.
• To classify representations in category O of type A rational Cherednik algebras that lack an S_n-invariant vector.
• To confirm a conjecture by Okounkov and the first author on the number of such representations.
• To establish the simplicity of the spherical Cherednik algebra of type A for parameters in the interval (−1, 0).

Proposed method

• The authors define parabolic induction and restriction functors within the framework of rational Cherednik algebras, generalizing standard constructions from Lie algebra representation theory.
• They apply these functors to analyze the structure of category O and the spherical subalgebra, particularly focusing on the role of S_n-invariant vectors.
• The proof of the Gordon-Stafford theorem is re-established without assuming c is not a half-integer, using the new functorial machinery.
• A classification of representations in category O without S_n-invariant vectors is achieved via characterizing their composition factors and using the induction/restriction functors.
• The simplicity of the spherical Cherednik algebra for −1 < c < 0 is proven using the structure of the category O and properties of the functors.
• An appendix by the second author includes results on the reducibility of the polynomial representation of the trigonometric Cherednik algebra, extending earlier work.

Experimental results

Research questions

• RQ1 How can parabolic induction and restriction functors be defined and applied in the context of rational Cherednik algebras?
• RQ2 Can the Gordon-Stafford Morita equivalence for type A rational Cherednik algebras be established without the restriction that c is not a half-integer?
• RQ3 What is the complete classification of representations in category O of type A rational Cherednik algebras that do not contain an S_n-invariant vector?
• RQ4 Does the conjecture by Okounkov and the first author on the number of such representations hold true?
• RQ5 Is the spherical Cherednik algebra of type A simple for parameters in the interval (−1, 0)?

Key findings

• The Gordon-Stafford Morita equivalence between the rational Cherednik algebra of type A and its spherical subalgebra is proven without requiring c ≠ half-integer.
• The paper classifies all representations in category O of type A rational Cherednik algebras that lack an S_n-invariant vector.
• The number of such representations is confirmed to match the prediction in the conjecture by Okounkov and the first author.
• The spherical Cherednik algebra of type A is shown to be simple for all parameters c satisfying −1 < c < 0.
• The polynomial representation of the trigonometric Cherednik algebra is proven to be reducible, as established in the appendix.
• The main results are strengthened and extended in the second version, including new insights into the structure of the category O and the role of the induction/restriction functors.