Reducibility of the polynomial representation of the degenerate double affine Hecke algebra

2007/06/28 by Pavel Etingof, Etingof, Pavel · 3 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT

paper · pdf · doi:10.48550/arxiv.0706.4308

7 pages, latex

arxiv created 2007/06/28 · openalex publication_date 2007/06/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we determine the values of parameters c for which the polynomial representation of the degenerate double affine Hecke algebra (DAHA), i.e. the trigonometric Cherednik algebra, is reducible. Namely, we show that c is a reducibility point for the polynomial representation of the trigonometric Cherednik algebra for a root system R if and only if it is a reducibility point for the rational Cherednik algebra for the Weyl group of some root subsystem R' of R of the same rank; such subsystems for any R are given by the well known Borel-de Siebenthal algorithm. This result has been proved by Cherednik using a case-by-case method.

Cited by

• A Lie-theoretic generalization of some Hilbert schemes
• Parabolic induction and restriction functors for rational Cherednik algebras
• Generalized Calogero-Moser systems from rational Cherednik algebras

Related

• Rational and trigonometric degeneration of the double affine Hecke algebra of type A
• Irreducibility of Perfect Representations of Double Affine Hecke Algebras
• The polynomial representation of the double affine Hecke algebra of type (C^\vee_n, C_n) for specialized parameters
• Derived equivalences for trigonometric double affine Hecke algebras
• Non-semisimple Macdonald polynomials