The quasiharmonic deformation conjecture for singular vectors

Let WW be a real reflection group with reflection representation VV, let H(W)H(W) be the universal rational Cherednik algebra, and let QHH(W)\mathcal{QH}\subset H(W) be the space of universal quasiharmonics. For a parameter cc, write M(c)(1)M^{(c)}(\mathbf{1}) for the polynomial representation, call a vector singular if it is annihilated by all Dunkl operators, and write evcQH\operatorname{ev}_c\mathcal{QH} for the specialization at cc. Quasiharmonic deformation conjecture. For every singular function cc, all the singular vectors of M(c)(1)M^{(c)}(\mathbf{1}) belong to

evcQH.\operatorname{ev}_c\mathcal{QH}.

Equivalently, the space of singular vectors in M(c)(1)M^{(c)}(\mathbf{1}) should be deformable within the class of quasiharmonics. The conjecture is proved for dihedral groups in the cases described by the source, while computations for symmetric groups support it; the general statement remains open.

Sources & referencesView supporting material

Primary source

Arkady Berenstein and Yurii Burman, “Quasiharmonic polynomials for Coxeter groups and representations of Cherednik algebras”, arXiv:math/0505173 (2007).

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